Computing and Visualization in Science

, Volume 17, Issue 1, pp 43–63 | Cite as

Convergence of goal-oriented adaptive finite element methods for semilinear problems

Article

Abstract

In this article we develop a convergence theory for goal-oriented adaptive finite element algorithms designed for a class of second-order semilinear elliptic equations. We briefly discuss the target problem class, and introduce several related approximate dual problems that are crucial to both the analysis as well as to the development of a practical numerical method. We then review some standard facts concerning conforming finite element discretization and error-estimate-driven adaptive finite element methods (AFEM). We include a brief summary of a priori estimates for this class of semilinear problems, and then describe some goal-oriented variations of the standard approach to AFEM. Following the recent approach of Mommer–Stevenson and Holst–Pollock for increasingly general linear problems, we first establish a quasi-error contraction result for the primal problem. We then develop some additional estimates that make it possible to establish contraction of the combined primal-dual quasi-error, and subsequently show convergence with respect to the quantity of interest. Finally, a sequence of numerical experiments are examined and it is observed that the behavior of the implementation follows the predictions of the theory.

Keywords

Adaptive finite element methods  Goal oriented Semilinear elliptic problems  Quasi-orthogonality Residual-based error estimator Convergence Contraction  A posteriori estimates 

Mathematics Subject Classification

65F08 65F10 65N30 65N50 65N55 

1 Introduction

In this article we develop convergence theory for a class of goal-oriented adaptive finite element methods for second order semilinear equations. In particular, we establish strong contraction results for a method of this type for the problem:with \(f \in L_2(\varOmega )\) and \(\varOmega \subset {\mathbb {R}}^d\) (\(d=2\) or 3) a polyhedral domain. We consider the problem with \(A:\varOmega \rightarrow \mathbb {R}^{d \times d}\) Lipschitz and symmetric positive definite (SPD). The standard weak formulation of the primal problem reads: Find \(u \in H^1_0(\varOmega )\) such that \(\forall v \in H_0^1(\varOmega )\):where
$$\begin{aligned} a(u,v) = \int _{\varOmega } A \nabla u \cdot \nabla v ~dx. \end{aligned}$$
(1.3)
In many practical applications, one is interested in certain physically relevant aspects of the solution, referred to as “quantities of interest”, such as (weighted) averages, flow rates or velocities. These quantities of interest are often characterized by the value g(u), where u is the solution of (1.1) and g is a linear functional associated with a particular “goal”. Given a numerical approximation \(u_{h}\) to the solution u, goal-oriented error estimates use duality techniques rather than the energy norm alone to estimate the error in the quantity of interest. The solution of the dual problem can be interpreted as the generalized Green’s function, or the influence function with respect to the linear functional, which often quantifies the stability properties of the computed solution. There is a substantial existing literature on developing reliable and accurate a posteriori error estimators for goal-oriented adaptivity; see [4, 6, 13, 14, 15, 19, 20, 32, 37] and the references cited therein. To our knowledge, the results presented here are the first to show convergence in the sense of the goal function for the class of semilinear elliptic problems discussed below. We support our theory with a numerical comparison of our method with standard goal-oriented adaptive strategies, demonstrating comparable efficiency with the added benefit of provable contraction for this problem class.

Our focus in this paper is on developing a goal-oriented adaptive algorithm for semilinear problems (1.2) along with a corresponding strong contraction result, following the recent approach in [24, 34] for linear problems. One of the main challenges in the nonlinear problem that we do not see in the linear case is the dependence of the dual problem on the primal solution u. As it is only practical to work with a dual problem we can accurately form, we develop a method for semilinear problems in which adaptive mesh refinement is driven both by residual-based approximation to the error in u, and by a sequence of approximate dual problems which depend on the latest numerical solution. While globally reducing the error in the primal problem alone eventually yields a good approximation to the quantity of interest g(u), it is not an efficient approach to the problem. The method we describe here refines the mesh with respect to both primal and dual problems at each iteration producing a sequence of increasingly accurate dual problems and corresponding influence functions to achieve an accurate approximation in fewer adaptive iterations; we emphasize the goal of the adaptive method is not to refine the fewest elements possible at each iteration, rather it is to target for refinement all dominant sources sources of error in the mesh. While adding elements to the primal refinement set may indeed lead to a less efficient method of approximating u, here it yields a more efficient process for approximating g(u) and achieves accuracy and efficiency comparable to other goal oriented methods with respect to mesh degrees of freedom.

Contraction of the adaptive finite element algorithm for the (primal) semilinear problem (1.2) has been established in [27] and [23]. Here we recall the contraction argument for the primal problem and use a generalization of this technique to establish the contraction of a linear combination of the primal and limiting dual quasi-errors by means of a computable sequence of approximate dual problems. We relate this result to a bound on the error in the quantity of interest. Following [27], the contraction argument follows from first establishing three preliminary results for two successive AFEM approximations \(u_1\) and \(u_2\), and respectively \(\hat{z}_1 \text { and } \hat{z}_2\) of the primal and limiting dual problems (see Sect. 2 for detailed definitions).
  1. 1.
    Quasi-orthogonality: There exists \( \varLambda _G > 1\) such that
     
  2. 2.
    Error estimator as upper bound on error: There exists \(C_1 > 0\) such that
     
  3. 3.
    Estimator reduction: For \({\mathcal {M}}\) the marked set that takes refinement \({\mathcal {T}}_1 \rightarrow {\mathcal {T}}_2\), for some constants \(\lambda \in (0,1)\), \(\varLambda _1>0\) and any \(\delta > 0\)
    $$\begin{aligned} \eta _2^2(v_2,{\mathcal {T}}_2) \le&(1 + \delta ) \{ \eta _1^2(v_1, {\mathcal {T}}_1) - \lambda \eta _1^2(v_1, {\mathcal {M}}) \} \\&+(1 + \delta ^{-1}) \varLambda _1 \eta _0^2 | \! | \! | {v_2 - v_1} | \! | \! |^2. \end{aligned}$$
     
For the primal problem, the mesh at each iteration may be marked for refinement with respect to the error indicators following the Dörfler marking strategy (cf. [12]). In the case of the dual problem, the limiting estimator as used in the contraction argument is related to a computable quantity. This quantity is the dual estimator, based on the residual of the approximate dual sequence. The mesh is marked for refinement with respect to this set of error indicators, which correspond to the approximate dual problem at each iteration. The transformation between limiting and approximate dual estimators couples the contraction of error in the limiting dual to the primal problem. The final result is the contraction of what we refer to here as the combined quasi-errorwhich is the sum of the quasi-error as in [9] for the limiting dual problem and a multiple of the quasi-error for the primal problem. The contraction of this property as shown in Theorem 5.1 establishes the contraction of the error in the goal function as shown in Corollary 5.3.

Our analysis is based on the recent development in the contraction framework for semilinear and more general nonlinear problems in [23, 25, 27], and those for linear problems developed by Cascon et al.  [9], and by Nochetto et al.  [36]. In addressing the goal-oriented problem we base our framework on that of Mommer and Stevenson [34] for symmetric linear problems and Holst and Pollock [24] for nonsymmetric problems. We note also two other recent convergence results in the literature for goal-oriented adaptive methods applied to self-adjoint linear problems, namely [11] and [35], both providing convergence rates in agreement with those in [34].

The analysis of the goal-oriented method for nonlinear problems is significantly more complex than the previous analysis for linear problems in [24, 34]. We follow a marking strategy similar to the one discussed in [24]; in particular, we mark for both primal and dual problems and take the union of the two as our marked set for the next refinement. This strategy differs from that in [34] in which they choose the set of lesser cardinality and use this to develop a quasi-optimal complexity result for solving Poisson’s equation. Due to the increased complexity of the problems we consider here, we show convergence with respect to the quasi-error as opposed to the energy error and as such mark for both primal and dual sets as the error estimator is not guaranteed to decrease monotonically for the dual problem if the mesh is only marked for the primal (and vice-versa). While we do not develop theoretical complexity results for this method, we demonstrate it efficiency numerically and see that it compares well with the method of [34] as well as the dual weighted residual (DWR) method. The analysis further departs from that in [24] as here we are faced with analyzing linearized and approximate dual sequences as opposed to a single dual problem in order to establish contraction with respect to the quantity of interest. The approach presented here allows us to establish a contraction result for the goal-oriented method, which appears to be the first result of this type for nonlinear problems.

Outline of the paper. The remainder of the paper is structured as follows. In Sect. 2, we introduce the approximate, linearized and limiting dual problems. We briefly discuss the problem class and review some standard facts concerning conforming finite element discretization and error-estimate-driven adaptive finite element methods (AFEM). In Sect. 2.2 we include a brief summary of a priori estimates for the semilinear problem. In Sect. 3, we describe a goal-oriented variation of the standard approach to AFEM (GOAFEM). In Sect. 4 we discuss contraction theorems for the primal problem. In Sect. 5 we introduce additional estimates necessary for the contraction of the combined quasi-error and convergence in the sense of the quantity of interest. Lastly, in Sect. 6 we present some numerical experiments that support our theoretical results.

2 Preliminaries

In this section, we state both the (nonlinear) primal problem and its finite element discretization. We then introduce the linearized dual problem, and consider some variants of this problem which are of use in the subsequent computation and analysis.

Consider the semilinear problem (1.2), where as in (1.3) we define the bilinear form
$$\begin{aligned} a(u, v) = ( A \nabla u, \nabla v ), \end{aligned}$$
with \(( \cdot , \cdot )\) denoting the \(L_2\) inner-product over \(\varOmega \subset {\mathbb {R}}^d\). We make the following assumptions on the data:

Assumption 1

The problem data \(\mathbf{D}= (A,b,f)\) satisfies
  1. 1)
    \(A: \varOmega \rightarrow {\mathbb {R}}^{d \times d}\) is Lipschitz continuous and symmetric positive-definite with
    $$\begin{aligned} \inf _{x \in \varOmega } \lambda _{\text {min}}(A(x))&= \mu _0 > 0,\\ \sup _{x \in \varOmega } \lambda _{\text {max}}(A(x))&= \mu _1 < \infty . \end{aligned}$$
     
  2. 2)
    \(b: \varOmega \times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is smooth on the second argument. Here and in the remainder of the paper, we write b(u) instead of b(xu) for simplicity. Moreover, we assume that b is monotone (increasing):
    $$\begin{aligned} b'(\xi ) \ge 0, ~\text { for all }\xi \in {\mathbb {R}}. \end{aligned}$$
     
  3. 3)

    \(f\in L_2(\varOmega )\).

     

Remark 2.1

Here we assumed that \(b(\cdot )\) satisfies the monotonicity Assumption 1(2) to simplify the presentation. However, all results in this paper can be easily generalized for the nonlinear problems with \(b'(\xi ) > -\lambda _1\) without any technical difficulty, where \(\lambda _1\) is the smallest eigenvalue of the diffusion operator \(-\nabla \cdot (A\nabla )\).

The native norm is the Sobolev \(H^1\) norm given by \(||{v} ||_{H^1}^2 = ( \nabla v, \nabla v ) + ( v,v ).\) Continuity of \(a(\, \cdot \, , \, \cdot \,)\) follows from the Hölder inequality,
$$\begin{aligned} a(u, v) \le \mu _1 |u|_{H^1} |v|_{H^1} \le M_{\mathcal {E}}||{u} ||_{H^1} ||{v} ||_{H^1}, \end{aligned}$$
(2.1)
where \(M_{\mathcal {E}}= \mu _{1}\). Define the energy semi-norm by the principal part of the differential operator Open image in new window. The coercivity of \(a(\, \cdot \, , \, \cdot \,)\) follows from the Poincaré inequality with constant \(C_\varOmega \)
$$\begin{aligned} a(v,v)&\ge \mu _0|v|_{H^1}^2 \ge C_\varOmega \mu _0 ||{v} ||_{H^1}^2 = m_{{\mathcal {E}}}^2||{v} ||_{H^1}^2, \end{aligned}$$
(2.2)
which establishes the energy semi-norm as a norm. Putting this together with (2.1) establishes the equivalence between the native and energy norms.

2.1 Linearized dual problems

Given a linear functional \(g\in L_{2}(\varOmega )\), the objective in goal-oriented error estimation is to relate the residual to the error in the quantity of interest. This involves solving a dual problem whose solution z satisfies the relation \(g(u - u_h) = \langle R(u_h), z \rangle \) where \(g(v):=\int _{\varOmega } gv dx\). In the linear case, the appropriate dual problem is the formal adjoint of the primal (cf. [24, 33]). For b nonlinear, the primal problem (1.2) does not have an exact formal adjoint. In this case we obtain the dual by linearization.

Formally, given a numerical approximation \(u_{j}\) to the exact solution u, the residual is given by
$$\begin{aligned} R(u_{j}):= f- {\mathcal {N}}(u_{j}) = {\mathcal {N}}(u) - {\mathcal {N}}(u_{j}). \end{aligned}$$
If \(z^{j}\in H_{0}^{1}(\varOmega )\) solves the following linearized dual problem
$$\begin{aligned} a(z^j,v) + \langle {\mathcal {B}}_j z^j,v\rangle&= g(v), \quad \forall v \in H_0^1(\varOmega ), \end{aligned}$$
(2.3)
where the operator \({\mathcal {B}}_j\) is given by
$$\begin{aligned} {\mathcal {B}}_j&\,{:=}\,\int _0^1 b'(\xi u + (1 - \xi ) u_j ) ~d \xi \nonumber \\&= \int _0^1 b'(u_j + (u - u_j)\xi ) ~d\xi , \end{aligned}$$
(2.4)
then the goal-oriented error \(g(e_j)\) of \(e_{j} = u - u_{j}\) can be represented exactly by the inner product of \(z^{j}\) and \(R(u_{j})\):
$$\begin{aligned} g(e_{j}) = \langle R(u_{j}), z^{j} \rangle . \end{aligned}$$
In fact, by definition of the residual \(R(u_{j})\), we have
$$\begin{aligned} \langle R(u_{j}), z^{j}\rangle= & {} a(z^{j}, e_{j}) + \langle z^{j}, b(u) - b(u_{j})\rangle \\= & {} a(z^{j}, e_{j}) + \langle {\mathcal {B}}_j z^j, e_j \rangle = g(e_{j}). \end{aligned}$$
Here we used the integral Taylor identity:
$$\begin{aligned} b(u)-b(u_{j})&= \int _0^1 b'(u_j + (u - u_j)\xi ) ~d\xi (u-u_{j}) \\&={\mathcal {B}}_{j} (u-u_{j}). \end{aligned}$$
The derivation and numerical use of the linearized dual problem is further discussed in [15, 16, 22].
Unfortunately, the dual problem (2.3) is not computationally useful because the operator \({\mathcal {B}}_{j}\) depends on the exact solution u. In order to define a computable dual operator, we introduce the approximate operator \(b'(u_j)\), which lead to the following approximate dual problem: Find \(\hat{z}^j \in H^1_0(\varOmega )\) such that
$$\begin{aligned} a(\hat{z}^j,v) + \langle b'(u_j) \hat{z}^j,v\rangle&= g(v), \quad \forall v \in H_0^1(\varOmega ). \end{aligned}$$
(2.5)
The Eq. (2.5) is instrumental for defining a computable a posteriori error indicator for the dual problem.
A further difficulty arises in the analysis of the goal-oriented adaptive algorithm driven by the a posteriori error estimators for the approximate dual problem (2.5). Due to the dependence on \(u_{j}\), (2.5) changes at each step of the adaptive algorithm. This is one of the essential differences of the nonlinear problem as compared to the linear cases in the previous literature (cf. [24, 34]). To handle this obstacle, we introduce the limiting dual problem: Find \(\hat{z} \in H^1_0(\varOmega )\) such that
$$\begin{aligned} a(\hat{z},v) + \langle b'(u) \hat{z}, v\rangle&=g(v), \quad \forall v \in H_0^1(\varOmega ). \end{aligned}$$
(2.6)
While the operator \(b'(u)\) is a function of the exact solution u and is not a computable quantity, it is the operator used in the limit of both the linearized dual (2.3) and approximate dual problems (2.5) as \(u_j \rightarrow u\). Therefore, both the linearized and approximate sequences approach the same limiting problem (2.6). Our contraction result in Theorem 5.1 is written with respect to the limiting dual problem as defined by the operator \(b'(u)\).

2.2 Finite element approximation

For a given conforming, shape-regular triangulation \({\mathcal {T}}\) of \(\varOmega \) consisting of closed simplices \(T\in {\mathcal {T}}\), we define the finite element space
$$\begin{aligned} {\mathbb {V}}_{\mathcal {T}}\,{:=}\,H_0^1(\varOmega ) \cap \prod _{T \in {\mathcal {T}}} {\mathbb {P}}_n(T) \quad \text { and } {\mathbb {V}}_k \,{:=}\,{\mathbb {V}}_{{\mathcal {T}}_k}, \end{aligned}$$
(2.7)
where \({\mathbb {P}}_n(T)\) is the space of polynomials degree \(\le n\) over T. For any subset \({\mathcal {S}}\subseteq {\mathcal {T}}\),
$$\begin{aligned} {\mathbb {V}}_{\mathcal {T}}({\mathcal {S}}) \,{:=}\,H_0^1(\varOmega ) \cap \prod _{T \in {\mathcal {S}}} {\mathbb {P}}_n(T). \end{aligned}$$
(2.8)
Given a triangulation \({\mathcal {T}}\), we denote \(h_{{\mathcal {T}}}:= \max _{T\in {\mathcal {T}}} h_{T}\) where \(h_{T} := |T|^{1/d}.\) In particular, we denote \(h_{0}:= h_{{\mathcal {T}}_{0}}\) for an initial (conforming, shape-regular) triangulation \({\mathcal {T}}_{0}\) of \(\varOmega \). Then the adaptive algorithm discussed below generates a nested sequence of conforming refinements \(\{{\mathcal {T}}_{k}\}\), with \({\mathcal {T}}_{k}\ge {\mathcal {T}}_{j}\) for \(k\ge j\) meaning that \({\mathcal {T}}_{k}\) is a conforming triangulation of \(\varOmega \) based on certain refinements of \({\mathcal {T}}_{j}\). With this notation, we also simply denote by \({\mathbb {V}}_{k}:= {\mathbb {V}}_{{\mathcal {T}}_{k}}\) the finite element space defined on \({\mathcal {T}}_{k}\).
The finite element approximation of the primal problem (1.2) reads: Find \(u_k \in {\mathbb {V}}_k \text { such that }\)
$$\begin{aligned} a(u_k, v_k) + \langle b(u_k), v_k \rangle = f(v_k), ~ v_k \in {\mathbb {V}}_k, \end{aligned}$$
(2.9)
and the finite element approximation of (2.5) linearized about \(u_j\) is given by: Find \(\hat{z}^j_k \in {\mathbb {V}}_k \) such that
$$\begin{aligned} a(\hat{z}^j_k,v_k) + \langle b'(u_j) \hat{z}^j_k, v_k \rangle = g(v_k) \quad \forall v_k \in {\mathbb {V}}_k. \end{aligned}$$
(2.10)
Finally, for the purpose of analysis, we introduce the discrete limiting dual problem (cf. (2.6)) given by: Find \(\hat{z}_k \in {\mathbb {V}}_k \) such that
$$\begin{aligned} a(\hat{z}_k,v_k) + \langle b'(u) \hat{z}_k, v_k \rangle = g(v_k) \quad \forall v_k \in {\mathbb {V}}_k. \end{aligned}$$
(2.11)
Existence and uniqueness of solutions to the primal problems (1.2) and (2.9) follow from standard variational or fixed-point arguments as in  [31] and  [40]. For the dual problems (2.5)–(2.6) and (2.10)–(2.11) the existence and uniqueness of solutions follow from the standard Lax-Milgram Theorem as in [18], since we assumed that \(b'(\xi ) \ge 0.\)

We make the following assumption on the a priori \(L_{\infty }\) bounds of the solutions to the primal problems (1.2) and (2.9):

Assumption 2

(A priori bounds) Let u and \(u_{k}\) be the solution to (1.2) and (2.9), respectively. We assume there are \(u_-, u_+ \in L_\infty \) which satisfy
$$\begin{aligned} u_-(x) \le u(x), u_k(x) \le u_+(x) ~\text {for a.e. } x \in \varOmega . \end{aligned}$$
(2.12)

Remark 2.2

The \(L_\infty \) bound on u follows from the standard maximum principle, as discussed in [3, Theorem 2.4] and [26, Theorem 2.3]. There is a significant literature on \(L_\infty \) bounds for the discrete solution, usually requiring additional angle conditions on the triangulation (cf. [26, 28, 29, 30] and the references cited therein). On the other hand, if b satisfies the (sub)critical growth condition, as stated in [3, Assumption (A4)], then the \(L_\infty \) bounds on the discrete solution \(u_k\) are satisfied without angle conditions on the mesh; see [3] for more detail.

Assumption 1 together with Assumption 2 yield the following properties on the continuous and discrete solutions as summarized below.

Proposition 1

Let the problem data satisfy Assumption 1 and Assumption 2. The following properties hold:
  1. 1)

    b is Lipschitz on \([u_-,u_+]\cap H_0^1(\varOmega )\) for a.e. \(x \in \varOmega \) with constant B.

     
  2. 2)

    \(b'\) is Lipschitz on \([u_-,u_+]\cap H_0^1(\varOmega )\) for a.e. \(x \in \varOmega \) with constant \(\varTheta \).

     
  3. 3)
    Let \(\hat{z}\) bet the solution to (2.6), \(\hat{z}_j^j\) the solution to (2.10) and \(\hat{z}_j\) the solution to (2.11). Then there are \(z_-, z_+ \in L_\infty \) which satisfy
    $$\begin{aligned} z_-(x) \le \hat{z}(x), \hat{z}_j(x), \hat{z}_j^j(x) \le z_+(x) ~\text {for a.e. } x \in \varOmega . \end{aligned}$$
    (2.13)
     

3 Goal oriented AFEM

In this section, we describe the goal oriented adaptive finite element method (GOAFEM), which is based on the standard AFEM algorithm:
$$\begin{aligned} \text { SOLVE } \rightarrow \text { ESTIMATE } \rightarrow \text { MARK } \rightarrow \text { REFINE }. \end{aligned}$$
(3.1)
Below, we explain each procedure.

Procedure SOLVE. The procedure SOLVE involves solving (2.9) for \(u_j\), computing \(b'(u_j)\) to form problem (2.10) and solving (2.10) for \(\hat{z}_j^j\). In the analysis that follows, we assume for simplicity the exact Galerkin solution is found on each mesh refinement. In practice the nonlinear problem (2.9) may be solved by a standard inexact Newton + multilevel algorithm as in [25]. The approximate dual problem (2.10) may be solved by any standard linear-time iterative method.

Procedure ESTIMATE. We use a standard residual-based element-wise error estimator for both primal and approximate dual problems. Recall the residual of the primal problem is given by Open image in new window with Open image in new window. For the limiting and approximate dual problems, we define the local strong form by Open image in new window, and \(\hat{{\mathcal {L}}}_j^*(v) \,{:=}\,-\nabla \cdot (A \nabla v) + b'(u_j)(v).\) The limiting and approximate dual residuals given respectively by
$$\begin{aligned} R^*(v) \,\,{:=}\,g - \hat{{\mathcal {L}}}^*(v), ~\text { and }~ \hat{R}_j^*(v) \,\,{:=}\,g - \hat{{\mathcal {L}}}_j^*(v). \end{aligned}$$
(3.2)
The jump residual for both the primal and linearized dual problems is:
$$\begin{aligned} J_T(v) \,\,{:=}\,[\![[A \nabla v] \cdot n ]\!]_{\partial T}, \end{aligned}$$
where \([\![~\cdot ~ ]\!]\) is given by
$$\begin{aligned}{}[\![\phi ]\!]_{\partial T} \,{:=}\,\lim _{t \rightarrow 0} \phi (x + t n) - \phi (x - tn) \end{aligned}$$
and n is taken to be the appropriate outward normal defined on \(\partial T\). For any \(v \in {\mathbb {V}}_{\mathcal {T}}\), the error indicator for the primal problem (2.9) is given by
$$\begin{aligned} \eta _{\mathcal {T}}^2(v,T) \,\,{:=}\,\, h_T^2 ||{R(v)} ||_{L_2(T)}^2 + h_T ||{ J_T(v) } ||_{L_2(\partial T)}^2. \end{aligned}$$
(3.3)
Similarly, for any \(w \in {\mathbb {V}}_{\mathcal {T}}\) the dual error-indicator is given by the approximate residual
$$\begin{aligned} \zeta _{{\mathcal {T}},j}^2(w,T) \,{:=}\,h_T^2 ||{\hat{R}_j^*( w)} ||_{L_2(T)}^2 + h_T ||{ J_T( w) } ||_{L_2(\partial T)}^2. \end{aligned}$$
(3.4)
This dual indicator is defined in terms of the approximate dual operator \(b'(u_j)\) as this is a computable quantity given an approximation \(u_j\). In addition, for purpose of analysis we define the limiting dual error-indicator for any \(w \in {\mathbb {V}}_{\mathcal {T}}\) by
$$\begin{aligned} \zeta _{{\mathcal {T}}}^2(w,T) \,\,{:=}\,\, h_T^2 ||{\hat{R}^*( w)} ||_{L_2(T)}^2 + h_T ||{ J_T( w) } ||_{L_2(\partial T)}^2. \end{aligned}$$
(3.5)
We remark the limiting dual indicator as given by (3.5) is not computable. For any given subset \({\mathcal {S}}\subset {\mathcal {T}}\), the error estimators on \({\mathcal {S}}\) are given by the \(l_2\) sum of error indicators over elements in the space.
$$\begin{aligned} \eta _{\mathcal {T}}^2(v, {\mathcal {S}}) \,\,{:=}\,\sum _{T \in {\mathcal {S}}} \eta _{{\mathcal {T}}}^2(v,T) , \quad v \in {\mathbb {V}}_{\mathcal {T}}, \end{aligned}$$
and similarly for the dual error estimator \(\zeta _{{\mathcal {T}},j}^2(w,{\mathcal {S}})\) and limiting estimator \(\zeta _{{\mathcal {T}}}^2(w,{\mathcal {S}})\). To simplify the notation, below we will omit “\({\mathcal {S}}\)” in the above definitions if \({\mathcal {S}}= {\mathcal {T}}\) and we will use \(\eta _{k}\) to denote \(\eta _{{\mathcal {T}}_{k}}\), and similarly use \(\zeta _{k,\cdot }\) to denote \(\zeta _{{\mathcal {T}}_{k}, \cdot }\).
As in [9] it is not difficult to verify the indicators for the primal and approximate (respectively limiting) dual problems satisfy the monotonicity property for \(v \in {\mathbb {V}}({\mathcal {T}}_1)\) and \({\mathcal {T}}_2 \ge {\mathcal {T}}_1\)
$$\begin{aligned} \eta _2(v,{\mathcal {T}}_2)\le & {} \eta _1(v, {\mathcal {T}}_1),\nonumber \\ \zeta _{2,j}(v,{\mathcal {T}}_2)\le & {} \zeta _{1,j}(v,{\mathcal {T}}_1),\nonumber \\ \zeta _{2}(v,{\mathcal {T}}_2)\le & {} \zeta _{1}(v,{\mathcal {T}}_1). \end{aligned}$$
(3.6)
For any element \(T \in {\mathcal {T}}_2 \cap {\mathcal {T}}_1\), it is obvious that
$$\begin{aligned} \eta _2(v,T)= & {} \eta _1(v, T), \nonumber \\ \zeta _{2,j}(v,T)= & {} \zeta _{1,j}(v,T),\nonumber \\ \zeta _{2}(v,T)= & {} \zeta _{1}(v,T). \end{aligned}$$
(3.7)
Procedure MARK. The Dörfler marking strategy for the goal-oriented problem is based on the following steps as in [34]:
  1. 1)
    Given \(\theta \in (0,1)\), mark sets for each of the primal and dual problems:
    • Mark a set \({\mathcal {M}}_p \subset {\mathcal {T}}_k\) such that
      $$\begin{aligned} \eta _k^2(u_k,{\mathcal {M}}_{p}) \ge \theta ^2 \eta _k^2(u_k, {\mathcal {T}}_k). \end{aligned}$$
      (3.8)
    • Mark a set \({\mathcal {M}}_d \subset {\mathcal {T}}_k\) such that
      $$\begin{aligned} \zeta _{k,k}^2(\hat{z}_k^k,{\mathcal {M}}_{d}) \ge \theta ^2 \zeta _{k,k}^2(\hat{z}_k^k, {\mathcal {T}}_k). \end{aligned}$$
      (3.9)
     
  2. 2)

    Let \( {\mathcal {M}}= {\mathcal {M}}_p \cup {\mathcal {M}}_d\) be the union of sets found for the primal and dual problems respectively.

     
As in  [24] the set \({\mathcal {M}}\) differs from that in [34], where the set of lesser cardinality between \({\mathcal {M}}_p \text { and } {\mathcal {M}}_d\) is used. We emphasize the necessity of the choice here to obtain strong contraction both in terms of the primal problem and the combined primal-dual system. For Poisson’s equation investigated in [34], the contracting quantity is the energy error, whereas here we develop contraction arguments for the quasi-error which combines the energy error with the error estimator. As the sequence of estimators for the primal (dual) problem based on the latest solution at each iteration is not necessarily monotone decreasing unless the primal (dual) problem has been refined for, we refine for both primal and dual problems at each iteration in order to force convergence of the quasi-error for both primal and dual problems. As seen in (3.9) the mesh is marked with respect to the dual indicators of the approximate-sequence solutions \(\hat{z}_k^k\) as these are computable quantities. Sets \({\mathcal {M}}_p \text { and } {\mathcal {M}}_d\) with optimal cardinality (up to a factor of 2) can be chosen in linear time  by binning the elements rather than performing a full sort [34].

Procedure REFINE. The refinement (including the completion) is performed according to newest vertex bisection which was first proposed in [38]. It has been proved the bisection procedure will preserve the shape-regularity of the initial triangulation \({\mathcal {T}}_{0}\). The complexity and other properties of this procedure are now well-understood (see for example [7] and the references cited therein), and will simply be exploited here.

4 Contraction for the primal problem

In this section, we discuss the contraction of the primal problem (1.2), recalling results from [27] and [26]. The contraction argument relies on three main convergence results, namely quasi-orthogonality, error-estimator as upper bound on error and estimator reduction. We include the analogous results here for the limiting dual problem when they are identical or nearly identical.

4.1 Quasi-orthogonality

Orthogonality in the energy-norm Open image in new window does not generally hold in the semilinear problem. We rely on the weaker quasi-orthogonality result to establish contraction of AFEM (GOAFEM). The proof of the quasi-orthogonality relies on the following \(L_{2}\)-lifting property.

Lemma 4.1

(\(L_2\)-lifting) Let the problem data satisfy Assumption 1 and Assumption 2. Let u be the exact solution to (1.2), and \(u_1 \in {\mathbb {V}}_1\) the Galerkin solution to (2.9). Let \(w\in H^{1+s}(\varOmega ) \cap H_0^1(\varOmega )\) for some \(0 < s \le 1\) be the solution to the dual problem: Find \(w\in H_{0}^{1}(\varOmega )\) such that
$$\begin{aligned} a(w,v) + \langle {\mathcal {B}}_1 w, v \rangle = \langle u - u_1, v \rangle , \quad v \in H_0^1(\varOmega ), \end{aligned}$$
(4.1)
where the operator \({\mathcal {B}}_{1}\) is defined by
$$\begin{aligned} {\mathcal {B}}_1 \,{:=}\,\int _0^1 b'(\xi u + (1 - \xi ) u_1 ) ~d \xi . \end{aligned}$$
As in [1, 10, 17] we assume the regularity
$$\begin{aligned} |w|_{H^{1+s}(\varOmega )} \le K_R ||{u-u_{1}} ||_{L_2(\varOmega )} \end{aligned}$$
(4.2)
based on the continuity of the coefficients \(a_{ij}\) and of \(b'(\cdot )\). Then

Proof

The proof follows the standard duality arguments in [1, 24] and [8], adapted for the semilinear problem. Let \({\mathcal {I}}^h: H_{0}^{1}(\varOmega ) \rightarrow {\mathbb {V}}_{1}\) be a quasi-interpolator, satisfying
$$\begin{aligned} ||{w - {\mathcal {I}}^h w} ||_{H^1}&\le C_{\mathcal {I}}h_{{\mathcal {T}}_1}^s |w|_{H^{1+s}} \end{aligned}$$
(4.4)
$$\begin{aligned} ||{w - {\mathcal {I}}^h w} ||_{L_2}&\le \hat{C}_{\mathcal {I}}h_{{\mathcal {T}}_1}^{1+s} |w|_{H^{1+s}}. \end{aligned}$$
(4.5)
as discussed in [1, 39] and [24].
Consider the linearized dual problem (4.1) with \(v = u - u_1 \in H_0^1(\varOmega )\) expressed in primal form
$$\begin{aligned} a(u- u_1,w) + \langle {\mathcal {B}}_1(u - u_1), w \rangle = ||{u - u_1} ||_{L_2}^2. \end{aligned}$$
(4.6)
By Galerkin orthogonality, for \({\mathcal {I}}^hw \in {\mathbb {V}}_1\)
$$\begin{aligned} a(u - u_1, {\mathcal {I}}^hw) + \langle {\mathcal {B}}_1(u - u_1), {\mathcal {I}}^h w\rangle = 0. \end{aligned}$$
(4.7)
Subtracting (4.7) from (4.6)
$$\begin{aligned} a(u\!-\! u_1,w\!-\! {\mathcal {I}}^hw) \!+\! \langle b(u) \!-\!b(u_1), w \!-\! {\mathcal {I}}^hw\rangle \!=\! ||{u \!-\! u_1} ||_{L_2}^2. \end{aligned}$$
(4.8)
Then by (2.1) continuity of \(a(\, \cdot \, , \, \cdot \,)\), the Hölder inequality and Lipschitz continuity of b (Proposition 1):
$$\begin{aligned} ||{u - u_1} ||_{L_2}^2 \le&M_{\mathcal {E}}||{u - u_1} ||_{H^1}||{w - {\mathcal {I}}^hw} ||_{H^1} \nonumber \\&+ B ||{u - u_1} ||_{L_2}||{w - {\mathcal {I}}^hw} ||_{L_2}. \end{aligned}$$
(4.9)
By coercivity (2.2), interpolation estimate (4.4), and regularity (4.2) on the first term on the RHS of (4.9)For the second term of (4.9), apply (4.5) followed by (4.2) and coercivity to the interpolation error yieldingApplying (4.10) and (4.11) to (4.9), we obtainThis completes the proof. \(\square \)

Similarly, we have the following \(L_{2}\)-lifting result for two Galerkin solutions.

Corollary 4.1

Let the assumptions in Lemma 4.1 hold. Let \(u_1 \in {\mathbb {V}}_1\) and \(u_2 \in {\mathbb {V}}_2\) be the Galerkin solutions to (2.9) in the spaces \({\mathbb {V}}_{1} \subset {\mathbb {V}}_{2}\), respectively. Then there is a constant \(C_{*}>0\) such that

Proof

The proof of (4.13) follows by replacing u by \(u_2\) in Lemma 4.1. In this case, we should replace the dual problem (4.1) by: Find \(w \in {\mathbb {V}}_2\) such that
$$\begin{aligned} a(w,v) + \langle {\mathcal {B}}_{12} w, v \rangle = ( u_2 - u_1, v ), \quad v \in {\mathbb {V}}_2, \end{aligned}$$
(4.14)
where the operator \({\mathcal {B}}_{12} \,{:=}\,\int _0^1 b'(\xi u_2 + (1 - \xi ) u_1 ) ~d \xi \). The rest of the proof is the same as Lemma 4.1. \(\square \)

Remark 4.1

As the dual problem (4.1) changes at each iteration, so may the regularity constant as given by (4.2) as well as the interpolation constants as given by (4.4) and (4.5). As such, the previous lemma shows a \(C_{*,k}\) for \(k = 1, 2, \ldots \). As the algorithm is run finitely many times, we consolidate these \(C_{*,k}\) into a single constant \(C_*\) for simplicity of presentation.

Now we are in position to show quasi-orthogonality.

Lemma 4.2

(Quasi-orthogonality) Let the problem data satisfy Assumption 1 and Assumption 2. Let \({\mathcal {T}}_1, {\mathcal {T}}_2\) be two conforming triangulation of \(\varOmega \) with \({\mathcal {T}}_2 \ge {\mathcal {T}}_1\). Let \(u\in H_0^1(\varOmega )\) be the exact solution to (1.2), \(u_i \in {\mathbb {V}}_i\) the solution to (2.9), \(i = 1,2\). There exists a constant \(C_*> 0\) depending on the problem data \(\mathbf{D}\) and initial mesh \({\mathcal {T}}_0\), and a number \(0 < s \le 1\) related to the angles of \(\partial \varOmega \), such that if the meshsize \(h_0\) of the initial mesh satisfies \(\bar{\varLambda } \,\,{:=}\,\, Bm_{\mathcal {E}}^{-1}C_*h_0^{s} < 1\), thenand in particular for \(\bar{v} = u_1\in {\mathbb {V}}_{1}\subset {\mathbb {V}}_{2}\)where
$$\begin{aligned} \varLambda \,\,{:=}\,\, (1 - B m_{\mathcal {E}}^{-1}C_*h_0^{s})^{-1} ~\text { and }~ \varLambda _G \,\,{:=}\,\, (1 - BC_*^2 h_0^{2s})^{-1} \end{aligned}$$
and \(C_*\) is the constant from Lemma 4.1.

Proof

For any given \(\bar{v} \in {\mathbb {V}}_{2}\), we haveBy Galerkin orthogonality
$$\begin{aligned} a(u - u_2, v) + \langle b(u) - b(u_2), v \rangle = 0 \text { for all }v \in {\mathbb {V}}_2, \end{aligned}$$
(4.18)
and taking \(v = \bar{v} - u_2\) in (4.18), we have
$$\begin{aligned} 2a(u - u_2, \bar{v} - u_2)&\le 2 |\langle b(u) - b(u_2), \bar{v} - u_2\rangle | \nonumber \\&\le 2B ||{u - u_2} ||_{L_2} ||{\bar{v} - u_2} ||_{L_2}. \end{aligned}$$
(4.19)
Here we used Hölder inequality and the Lipschitz property on b (cf. Proposition 1).
To prove the inequality (4.15), by applying the \(L_2\)-lifting Lemma 4.1 to the first factor on the RHS and the coercivity (2.2) to the second followed by Young’s inequality, we obtainApplying (4.20) via (4.19) to (4.17)Assuming \( \bar{\varLambda } \,\,{:=}\,\, Bm_{\mathcal {E}}^{-1}C_*h_0^{s} < 1, \) we havewith \(\varLambda = (1 - Bm_{\mathcal {E}}^{-1}C_*h_0^{s})^{-1}\).
The proof of the inequality (4.16) is almost identical. By applying \(L_2\)-lifting 4.1 to each norm on the RHS of (4.19) by means of Corollary 4.1 then applying Young’s inequalityFollowing the same procedure as above yieldswith \(\varLambda _G = (1 - BC_*^2 h_0^{2s})^{-1}\) with the weaker mesh assumption \(\bar{\varLambda }_G \,{:=}\,BC_*^2h_0^{2s}<1\). \(\square \)

We note the second Galerkin orthogonality estimate (4.23) sharpens our results but is not essential to establishing them.

4.2 Error estimator as global upper-bound

The second key result for the contraction of the primal problem is the error estimator as a global upper bound on the energy error, up to a global constant. The result for the semilinear problem is established in [23, 27] with a clear generalization to the approximate dual sequence, also see [9] and [33] for the linear cases. The proof of this result follows from the general a posteriori error estimation framework developed in [41, 42].

Lemma 4.3

(Error estimator as global upper-bound) Let the problem data satisfy Assumption 1 and Assumption 2. Let \({\mathcal {T}}_{k}\) be a conforming refinement of \({\mathcal {T}}_{0}\). Let \(u\in H_{0}^{1}(\varOmega )\) and \(u_k \in {\mathbb {V}}_k\) be the solutions to (1.2) and (2.9), respectively. Similarly, let \(\hat{z} \in H_{0}^{1}(\varOmega )\) and \(\hat{z}_k \in {\mathbb {V}}_k\) be the solutions to (2.6) and (2.11), respectively. Then there is a global constant \(C_1\) depending only on the problem data \(\mathbf{D}\) and initial mesh \({\mathcal {T}}_0\) such thatand

4.3 Estimator reduction

The local Lipschitz property as in [27], analogous to the local perturbation property established in [9], is a key step in establishing estimator reduction leading to the contraction result. For any \(T\in {\mathcal {T}}\), we denote
$$\begin{aligned} \omega _T \,\,{:=}\,\, T \cup \{ T' \in {\mathcal {T}}~\big |~ T \cap T' \text { is a true-hyperface of } T \}.\nonumber \\ \end{aligned}$$
(4.26)
Here, for a d-simplex T, a true-hyperface is a \(d-1\) sub-simplex of T, e.g., a face in 3D or an edge in 2D. We also define the data estimator on each element \(T \in {\mathcal {T}}\) as
$$\begin{aligned} \eta _{\mathcal {T}}^2(\mathbf{D},T) = h_T^2 \left( ||{{{\text {div}}}A} ||_{L_\infty (T)}^2 + h_T^{-2 } ||{A} ||_{L_\infty (\omega _T)}^2 + B^2 \right) , \end{aligned}$$
(4.27)
and denote \( \eta _{\mathcal {T}}(\mathbf{D},{\mathcal {S}}) = \max _{T \in {\mathcal {S}}} \eta _{{\mathcal {T}}}(\mathbf{D},T) \) for any subset \({\mathcal {S}}\subseteq {\mathcal {T}}\). Recall that B is the Lipschitz constant in Proposition  1. In particular, we denote by \( \eta _0 \,{:=}\,\eta _{{\mathcal {T}}_0} (\mathbf{D},{\mathcal {T}}_0) \) the data estimator on the initial mesh. As the grid is refined, the data estimator satisfies the monotonicity property for refinements \({\mathcal {T}}_2 \ge {\mathcal {T}}_1\) (cf. [9]):
$$\begin{aligned} \eta _2(\mathbf{D},{\mathcal {T}}_2) \le \eta _1(\mathbf{D},{\mathcal {T}}_1). \end{aligned}$$
(4.28)

Lemma 4.4

(Local Lipschitz Property) Let the problem data satisfy Assumption 1 and Assumption 2. Let \({\mathcal {T}}\) be a conforming refinement of \({\mathcal {T}}_{0}\). Then for all \(T \in {\mathcal {T}}\) and for any \(v, w \in {\mathbb {V}}_{\mathcal {T}}\)
$$\begin{aligned} |\eta _{\mathcal {T}}(v,T) - \eta _{\mathcal {T}}(w,T)| \le \bar{\varLambda }_1 \eta _{\mathcal {T}}(\mathbf{D},T)||{v - w} ||_{H^1(\omega _T)}. \end{aligned}$$
(4.29)
The constant \(\bar{\varLambda }_1> 0\) depends on the dimension d and the initial mesh \({\mathcal {T}}_0\).

Proof

The proof follows those in [9] and [24], and we sketch the proof below. From (3.3), for any \(v \in {\mathbb {V}}_{\mathcal {T}}\) we have
$$\begin{aligned} \eta _{\mathcal {T}}^2(v,T) \,\,{:=}\,\, h_T^2 ||{R(v)} ||_{L_2(T)}^2 + h_T ||{ J_T(v) } ||_{L_2(\partial T)}^2. \end{aligned}$$
(4.30)
Set \(e = v - w\) and by definition of the residual, we get
$$\begin{aligned} R(v) =&f - {\mathcal {N}}( w + e) \\ =&f + \nabla \cdot (A \nabla w) - b(w) \\&+\nabla \cdot (A \nabla e) - \left( \int _0^1 b'(w + \xi e) ~d\xi \right) e \\ =&R(w) + {\mathcal {D}}(e), \end{aligned}$$
where \({\mathcal {D}}(e) \,{:=}\,\nabla \cdot (A \nabla e) - \left( \int _0^1 b'(w + \xi e) ~d\xi \right) e\). Using the generalized triangle-inequality for any \(a,b,c,d > 0\):
$$\begin{aligned} \sqrt{(a + b)^2 + (c + d)^2} \le \sqrt{a^2 + c^2} + b + d, \end{aligned}$$
and linearity of the jump residual we have
$$\begin{aligned} \eta _{{\mathcal {T}}}(v,T) =&\left( h_T^2 ||{R(w) + {\mathcal {D}}(e)} ||_{L_2(T)}^2 \right. \nonumber \\&\left. +h_T ||{J(w) + J(e)} ||_{L_2(\partial T)}^2\right) ^{1/2} \nonumber \\ \le \eta _{{\mathcal {T}}}(w,T)&+ h_T||{{\mathcal {D}}(e)} ||_{L_2(T)} + h_T^{\scriptstyle 1/2}||{J(e)} ||_{L_2(\partial T)}. \end{aligned}$$
(4.31)
For the second term of (4.31), by triangle inequality we obtain
$$\begin{aligned} ||{{\mathcal {D}}(e)} ||_{L_2(T)} \le&||{\nabla \cdot (A \nabla e)} ||_{L_2(T)} \nonumber \\&+\left\| {\left( \int _0^1 b'(w + \xi e) ~d\xi \right) e } \right\| _{L_2(T)}. \end{aligned}$$
(4.32)
By the inverse inequality, the diffusion term satisfies the bound
$$\begin{aligned}&||{\nabla \cdot (A \nabla e)} ||_{L_2(T)} \end{aligned}$$
(4.33)
$$\begin{aligned}&\quad \le ||{{{\text {div}}}A \cdot \nabla e} ||_{L_2(T)} + ||{A : D^2 e} ||_{L_2(T)} \nonumber \\&\quad \le \left( ||{{{\text {div}}}A} ||_{L_\infty (T)} + C_I h_T^{-1 } ||{A} ||_{L_\infty (T)} \right) ||{\nabla e} ||_{L_2(T)}, \end{aligned}$$
(4.34)
where \(D^{2} e\) is the Hessian of e. The second term in  (4.32) is bounded by
$$\begin{aligned} \left\| { \left( \int _0^1 b'(w + \xi e) ~d\xi \right) e } \right\| _{L_2(T)} \le B ||{e} ||_{L_2(T)}. \end{aligned}$$
(4.35)
The jump term in (4.31) satisfies
$$\begin{aligned}&||{J(e)} ||_{L_2(\partial T)}\nonumber \\&\quad \le 2 (d+1)~ C_T h_{T}^{-{\scriptstyle 1/2}} ||{A} ||_{L_\infty (\omega _T)} ||{\nabla e} ||_{L_2(\omega _T)}\nonumber \\&= C_Jh_{T}^{-{\scriptstyle 1/2}} ||{A} ||_{L_\infty (\omega _T)} ||{\nabla e} ||_{L_2(\omega _T)}, \end{aligned}$$
(4.36)
where \(C_T\) depends only on the shape-regularity of the triangulation. Putting together (4.31), (4.33), (4.35) and (4.36), we obtain
$$\begin{aligned} \eta _{{\mathcal {T}}}(v,T)&\le \eta _{{\mathcal {T}}}(w,T) + h_T \left( ||{{{\text {div}}}A} ||_{L_\infty (T)} \right. \nonumber \\&+ \left. (C_I + C_J) h_T^{-1 } ||{A} ||_{L_\infty (\omega _T)} + B \right) ||{ e} ||_{H^1(\omega _T)}\nonumber \\&\le \eta _{{\mathcal {T}}}(w,T) + C_{TOT}\, \eta _{\mathcal {T}}(\mathbf{D},T)||{v - w} ||_{H^1(\omega _T)}. \end{aligned}$$
(4.37)
This completes the proof. \(\square \)

The local perturbation property demonstrated in Lemma 4.4 (respectively, Lemma 5.3 below) leads to estimator reduction, one of the three key ingredients for contraction of the both the primal and combined quasi-errors. This result holds for both the primal and limiting dual problems, whose proof can be found in [9, Corollary 2.4] or [24, Theorem 3.4].

Theorem 4.1

(Estimator reduction) Let the problem data satisfy Assumption 1 and Assumption 2. Let \({\mathcal {T}}_1\) be a conforming refinements of \({\mathcal {T}}_{0}\), \({\mathcal {M}}\subset {\mathcal {T}}_1\) be the marked set, and \({\mathcal {T}}_2 = \text {REFINE}({\mathcal {T}}_1, {\mathcal {M}})\). Let
$$\begin{aligned} \varLambda _1 \,\,{:=}\,\, (d+2)\bar{\varLambda }_1^2 m_{\mathcal {E}}^{-2} \quad \text { and }~ \lambda \,\,{:=}\,\, 1 - 2^{-1/d}> 0 \end{aligned}$$
with \(\bar{\varLambda }_1\) from Lemma 4.4 (local Lipschitz property). Then for any \(v_1 \in {\mathbb {V}}_1\) and \(v_2 \in {\mathbb {V}}_2\) and \(\delta > 0\)Analogously for the limiting dual problem

The contraction of the primal (semilinear) problem is established in [27] and [23] based on Lemma 4.2, Lemma 4.3 and Theorem 4.1 as discussed above.

Theorem 4.2

(Contraction of the primal problem) Let the problem data satisfy Assumption 1 and Assumption 2, and let the meshsize \(h_0\) satisfy the condition of Lemma 4.2. Let u the solution to (1.2). Let \(\theta \in (0,1]\), and let \(\{{\mathcal {T}}_j, {\mathbb {V}}_j, u_j\}_{j \ge 0}\) be the sequence of meshes, finite element spaces and discrete solutions produced by GOAFEM. Then there exist constants \(\gamma _p > 0 \text { and } 0 < \alpha < 1\), depending on the initial mesh \({\mathcal {T}}_0\) and marking parameter \(\theta \) such that

5 Contraction and convergence of GOAFEM

In this section, we discuss the contraction and convergence of the GOAFEM described in Sect. 3. In particular, we show the GOAFEM algorithm generates a sequence \(\{{\mathcal {T}}_j, {\mathbb {V}}_j, u_j, \hat{z}_{j}\}_{j}\) which contracts not only in the primal error as shown in Sect. 4, but also in a linear combination of the primal and limiting dual error. We emphasize that it would be difficult to derive convergence results in terms of problem (2.3) or (2.5), because at each refinement the problem is changing. So, we show contraction in terms of the error in the limiting dual problem (2.6) as the target equation is fixed over the entire adaptive algorithm. Our approach of showing contraction in this section again relies on three main components: quasi-orthogonality, error-estimator as upper bound on error and estimator reduction. Here we discuss the relevant results for the limiting dual problem with an emphasis on those that differ significantly from the corresponding results for the primal problem. Note the limiting dual problem is not computable: we connect the error for the limiting dual problem to the computable quantities in the GOAFEM algorithm. For this purpose, we introduce Lemma 5.4, converting between limiting and approximate estimators in order to apply the Dörfler property to a computable quantity; and Lemma 5.5, bounding the discrete error between approximate and limiting dual solutions in terms of the primal error. We put these results together in Theorem 5.1 to establish the contraction of the combined quasi-error. Finally, the contraction of this form of the error is related to the error in the quantity of interest in Corollary 5.3.

5.1 Quasi-orthogonality for limiting-dual problem

Similar to the proof of the quasi-orthogonality for the primal problem, we make use of an \(L_{2}\)-lifting argument for the limiting-dual problem. Let \(\hat{z} \in H_0^1(\varOmega )\) and \(\hat{z}_1 \in {\mathbb {V}}_1\) be the solutions to  (2.6) and (2.11), respectively. We again use the duality argument, and introduce the problem: Find \(y \in H_0^1(\varOmega )\) such that
$$\begin{aligned} a(y,v) + \langle b'(u) y, v\rangle = (\hat{z} - \hat{z}_{1}, v) \text { for all }v \in H_0^1(\varOmega ) \end{aligned}$$
(5.1)
Then we have the following \(L_{2}\)-lifting result for the limiting-dual problem.

Lemma 5.1

(Limiting-dual \(L_2\)-lifting) Let the problem data satisfy Assumption 1 and Assumption 2. Let \({\mathcal {T}}_1\) be a conforming triangulation, and \(\hat{z} \in H_0^1(\varOmega )\) and \(\hat{z}_1 \in {\mathbb {V}}_1\) be the solutions to (2.6) and (2.11), respectively. Assume the solution y to (5.1) belongs to \(H^{1+s}(\varOmega ) \cap H_0^1(\varOmega )\) for some \(0 < s \le 1\) such that
$$\begin{aligned} |y|_{H^{1+s}(\varOmega )} \le \bar{K}_R ||{\hat{z} - \hat{z}_1} ||_{L_2(\varOmega )}. \end{aligned}$$
(5.2)
Then

Proof

The proof is essentially the same as that of Lemma 4.1, we omit it here. \(\square \)

Remark 5.1

Similar to Corollary 4.1, the \(L_2\)-lifting Lemma 5.1 also holds for two Galerkin solutions to (2.11), \(\hat{z}_2 \in {\mathbb {V}}_2\) and \(\hat{z}_1\in V_1\) with \({\mathbb {V}}_{1} \subset {\mathbb {V}}_{2}\), namely,The proof is essentially the same. We only need to replace  (5.1) by the problem: Find \(y \in {\mathbb {V}}_2\) such that
$$\begin{aligned} a(y,v) + \langle b'(u) y, v\rangle = \langle \hat{z}_2 - \hat{z}_{1}, v\rangle \text { for all }v \in {\mathbb {V}}_2. \end{aligned}$$

With the help of Lemma 5.1, we obtain the following quasi-orthogonality for the limiting-dual problem.

Lemma 5.2

(Quasi-orthogonality for Limiting Dual Problem) Let the problem data satisfy Assumption 1, and \({\mathcal {T}}_1, {\mathcal {T}}_2 \) be two conforming triangulations with \({\mathcal {T}}_2 \ge {\mathcal {T}}_1\). Let \(\hat{z} \in H_0^1(\varOmega )\) the solution to  (2.6) and \(\hat{z}_i \in {\mathbb {V}}_i\) the solution to (2.11), \(i = 1,2\). There exists a constant \(\hat{C}_*> 0\) depending on the problem data \(\mathbf{D}\) and initial mesh \({\mathcal {T}}_0\), and a number \(0 < s \le 1\) related to the regularity of (5.1), such that for sufficiently small \(h_0\) we haveand in particular for \(\bar{v} = \hat{z}_1\)where
$$\begin{aligned} \hat{\varLambda } \,\,{:=}\,\, (1 -B m_{\mathcal {E}}^{-1}\hat{C}_*h_0^{s})^{-1} ~\text { and }~ \hat{\varLambda }_G \,\,{:=}\,\, (1 - B \hat{C}_*^2 h_0^{2s})^{-1} \end{aligned}$$
and \(\hat{C}_*\) is the constant from Lemma 5.1.

Proof

The proof follows same arguments as in Lemma 4.2, except that in place of the inequality in (4.18) we have for the limiting dual problem
$$\begin{aligned} a(u - u_2, v) + \langle b'(u)(\hat{z} - \hat{z}_2), v \rangle = 0 \forall v \in {\mathbb {V}}_2, \end{aligned}$$
(5.6)
yielding
$$\begin{aligned} 2a (\hat{z} - \hat{z}_2, \bar{v} -\hat{z}_2)&\le 2B ||{\hat{z} - \hat{z}_2} ||_{L_2} ||{\bar{v} - \hat{z}_2} ||_{L_2}, \end{aligned}$$
(5.7)
as in (4.19). The rest of the proof is similar to Lemma 4.2, and we omit it here. \(\square \)

5.2 Estimator perturbations for dual sequence

As we have seen in Theorem 4.1, the local Lipschitz property (cf. Lemma 4.4) plays a key role in deriving the estimator reduction property used to convert between estimators on different refinement levels in both the primal and limiting dual problems. The following lemma gives similar local Lipschitz properties for the approximate and limiting dual problems on a given refinement level.

Lemma 5.3

(Local Lipschitz Property for Dual Estimators) Let the problem data satisfy Assumption 1 and Assumption 2. Let \({\mathcal {T}}\) be a conforming refinement of \({\mathcal {T}}_{0}\). Then for all \(T \in {\mathcal {T}}\) and for any \(v, w \in {\mathbb {V}}_{\mathcal {T}}\), it holds that
$$\begin{aligned} |\zeta _{{\mathcal {T}},j}(v,T) - \zeta _{{\mathcal {T}},j}(w,T) |&\le \bar{\varLambda }_1 \eta _{\mathcal {T}}(\mathbf{D},T)||{v - w} ||_{H^1(\omega _T)}. \end{aligned}$$
(5.8)
In particular, for the error indicator of the limiting dual problem we have
$$\begin{aligned} |\zeta _{{\mathcal {T}}}(v,T) - \zeta _{{\mathcal {T}}}(w,T)|&\le \bar{\varLambda }_1 \eta _{\mathcal {T}}(\mathbf{D},T)||{v - w} ||_{H^1(\omega _T)}. \end{aligned}$$
(5.9)
The constant \(\bar{\varLambda }_1> 0\) depends on the dimension d and the regularity of the initial mesh \({\mathcal {T}}_0\).

Proof

The proof is similar to Lemma 4.4. We sketch the proof below. To prove (5.8), by (3.4) we have for any \(v \in {\mathbb {V}}_{\mathcal {T}}\):
$$\begin{aligned} \zeta _{{\mathcal {T}},j}^2(v,T) \,\,{:=}\,\, h_T^2 ||{\hat{R}_j^*(v)} ||_{L_2(T)}^2 + h_T ||{ J_T(v) } ||_{L_2(\partial T)}^2. \end{aligned}$$
(5.10)
Setting \(e = v - w\) and applying linearity to the definition of the dual residual as given by (3.2), we obtain
$$\begin{aligned} \hat{R}_j^*(v) = g + \hat{{\mathcal {L}}}_j^*( w + e) = \hat{R}_j^*(w) + \hat{{\mathcal {L}}}_j^*(e). \end{aligned}$$
By the same reasoning as (4.31), we get
$$\begin{aligned} \zeta _{{\mathcal {T}},j}(v,T) \le&\zeta _{{\mathcal {T}},j}(w,T) + h_T||{\hat{{\mathcal {L}}}_j^*(e)} ||_{L_2(T)} \nonumber \\&+ h_T^{\scriptstyle 1/2}||{J(e)} ||_{L_2(\partial T)}. \end{aligned}$$
(5.11)
The term \(\hat{{\mathcal {L}}}_j^*\) (respectively \(\hat{{\mathcal {L}}}^*\) for the limiting dual) in (5.11) satisfies the same bound as the analogous term \({\mathcal {D}}\) in  (4.31) of Lemma 4.4. Hence the bounds (5.8) and (5.9) hold with the same constants as in (4.29). \(\square \)

With the help of Lemma 5.3, we are able to derive the following corollary, which addresses the error induced by switching between error indicators corresponding to the approximate and limiting dual problems on a given element.

Corollary 5.1

Let the problem data satisfy Assumption 1 and Assumption 2. Let \({\mathcal {T}}\) be a conforming refinement of \({\mathcal {T}}_{0}\), and \(u, u_{j}\) are the solutions to (1.2) and (2.9) problems, respectively. Let \(\varTheta \text { and } K_Z\) the constants given in Proposition 1. For all \(T \in {\mathcal {T}}\) and for \(v,w \in {\mathbb {V}}_{\mathcal {T}}\cap [z_-,z_+]\) the dual indicator on \({\mathcal {T}}\) satisfies
$$\begin{aligned}&|\zeta _{{\mathcal {T}},j}(v,T) - \zeta _{{\mathcal {T}},k}(w,T)|\le \bar{\varLambda }_1 \eta _{\mathcal {T}}(\mathbf{D},T)||{v - w} ||_{H^1(\omega _T)}\nonumber \\&\quad + \varTheta K_Z h_T||{u_j - u_k} ||_{L_2(T)}. \end{aligned}$$
(5.12)
In particular, for \({\mathcal {T}}= {\mathcal {T}}_1\), we have for the limiting estimator
$$\begin{aligned} |\zeta _{1,1}(v,T) - \zeta _{1}(w,T)| \le&\bar{\varLambda }_1 \eta _1(\mathbf{D},T)||{v - w} ||_{H^1(\omega _T)}\nonumber \\&+ \varTheta K_Z h_T||{u - u_1} ||_{L_2(T)}. \end{aligned}$$
(5.13)

Proof

By the definition of the residuals for the approximate dual problems, for any \(w\in {\mathbb {V}}_{{\mathcal {T}}}\) we have
$$\begin{aligned} \hat{R}^*_j(w)&= g \!+\! \nabla \cdot (A \nabla w) + b'(u_k) w + \left( b'(u_j) - b'(u_k) \right) w\nonumber \\&= \hat{R}^*_k(w) + \left( b'(u_j) - b'(u_k) \right) w. \end{aligned}$$
(5.14)
Using (5.14) in the definition of the dual indicator (3.4) and applying a generalized triangle inequality
$$\begin{aligned} \zeta _{{\mathcal {T}},j}(w,T) =&\left( h_T^2 ||{\hat{R}^*_k(w) + (b'(u_j) - b'(u_k)) w} ||_{L_2(T)}^2 \right. \nonumber \\&\left. + h_T||{J_T(w)} || _{L_2(\partial T)}^2 \right) ^{1/2}\nonumber \\ \le&\left( h_T^2 ||{\hat{R}_k^*(w)} ||_{L_2(T)}^2 + h_T ||{J_T(w)} ||_{L_2(\partial T)}^2 \right) ^{1/2} \nonumber \\&+h_T ||{\left( b'(u_j) - b'(u_k)\right) w} ||_{L_2(T)} \end{aligned}$$
(5.15)
$$\begin{aligned}&\le \zeta _{{\mathcal {T}},k}(w,T) + \varTheta K_Z h_T||{u_j - u_k} ||_{L_2(T)}. \end{aligned}$$
(5.16)
Applying (5.8) in Lemma 5.3 to the estimate (5.15), we obtain (5.12). \(\square \)

As an immediate consequence of Corollary 5.1, we have the following results on the error induced by switching between dual estimators over a collection of elements on a given refinement level. This estimate plays a key role in the contraction argument below, as we apply it to switching between the estimator for the limiting dual and the computed error estimators for the approximate dual problems in the GOAFEM algorithm.

Corollary 5.2

Let the hypotheses of Corollary 5.1 hold. Then for any subsets \({\mathcal {M}}_1, {\mathcal {M}}_2 \subseteq {\mathcal {T}}_1\) and arbitrary positive constants \(\delta _1, \delta _2, \delta _A, \delta _B\)
$$\begin{aligned} \zeta _1^2(v,{\mathcal {M}}_1)&\ge (1 + \delta _1)^{-1} (1 + \delta _A)^{-1}\zeta _{1,1}^2(w,{\mathcal {M}}_1)\nonumber \\&\quad - (1 + \delta _1)^{-1} \delta _A^{-1} \varTheta ^2 K_Z^2 h_0^2||{u - u_1} ||_{L_2}^2 \nonumber \\&\quad -(d+2)\delta _1^{-1} \bar{\varLambda }_1^2 \eta _0^2 ||{v - w} ||_{H^1}^2 \end{aligned}$$
(5.17)
$$\begin{aligned} \zeta _{1,1}^2(w,{\mathcal {M}}_2)&\ge (1 + \delta _2)^{-1} (1 + \delta _B)^{-1}\zeta _{1}^2(v,{\mathcal {M}}_2)\nonumber \\&\quad - (1 + \delta _2)^{-1} \delta _B^{-1} \varTheta ^2 K_Z^2 h_0^2||{u - u_1} ||_{L_2}^2 \nonumber \\&\quad -(d+2)\delta _2^{-1} \bar{\varLambda }_1^2 \eta _0^2 ||{v - w} ||_{H^1}^2. \end{aligned}$$
(5.18)

Proof

The conclusions follow by squaring inequality (5.13), applying Young’s inequality twice, and then summing over element \(T \in {\mathcal {M}}_{1}\) (respectively \(T\in {\mathcal {M}}_{2}\)). The \(H^1\) norm is summed over all elements \(T \in {\mathcal {T}}_1\) counting each element \(d+2\) times, the maximum number of elements in each patch \(\omega _T\). \(\square \)

5.3 Contraction of GOAFEM

The main contraction argument Theorem 5.1 follows after two more lemmas. The first combines a sequence of estimates to convert the non-computable limiting estimator for the dual problem to a computable quantity, apply the Dörfler property and then convert back. The second relates the difference between the Galerkin solutions of the limiting and approximate dual problems to the primal error. Motivated by estimator reduction for the limiting dual problem as in equation (4.39)the following lemma addresses the conversion between the limiting estimator \(\zeta _{1}^2(\hat{z}_1,{\mathcal {M}})\) and and the computable estimator \(\zeta _{1,1}^2(\hat{z}_1^1,{\mathcal {M}})\) necessary for marking the mesh for refinement.

Lemma 5.4

Let the problem data satisfy Assumption 1 and Assumption 2. Let \(\varTheta \text { and } K_Z\) as given by Proposition 1, \(C_*\) as given by Lemma 4.1 and \(\varLambda _1\) as given in Lemma 4.1. Let
$$\begin{aligned} u&\text { the solution to}~(1.2),&u_1&\text { the solution to}~(2.9), \\ \hat{z}&\text { the solution to}~(2.6),&\hat{z}_1&\text { the solution to}~(2.11),\\ \hat{z}_1^1&\text { the solution to}~(2.10).&\end{aligned}$$
Let \(\zeta _{1,1}(\hat{z}_1^1, {\mathcal {M}})\) satisfy the Dörfler property (3.9) for \({\mathcal {M}}\subset {\mathcal {T}}_1\), namely \( \zeta _{1,1}^2(\hat{z}_1^1, {\mathcal {M}}) \ge \theta ^2 \zeta ^2_{1,1}(\hat{z}_1^1, {\mathcal {T}}_1). \) Then for arbitrary \(\delta _1, \delta _2, \delta _A, \delta _B > 0\) there is a \(\delta _4>0\) such that

Proof

From Corollary 5.2, \(L_2\)-lifting Lemma 4.1 and coercivity (2.2)with \(\varLambda _1 \,\,{:=}\,\, \bar{\varLambda }_1^2(d+2)m_{\mathcal {E}}^{-2}\). The Dörfler property may be applied to the first term on the RHS of (5.21)
$$\begin{aligned} - \zeta _{1,1}^2(\hat{z}_1^1,{\mathcal {M}}) \le - \theta ^2 \zeta _{1,1}^2(\hat{z}_1^1). \end{aligned}$$
(5.22)
Converting back to he limiting estimator by (5.18) in Corollary 5.2Define \(\delta _4\) by
$$\begin{aligned} (1 + \delta _4) \,\,{:=}\,\, (1 + \delta _1)(1 + \delta _2)(1 + \delta _A)(1 + \delta _B). \end{aligned}$$
(5.24)
Then by plugging (5.22) and (5.23) in the first term on the RHS of (5.21), we obtainFinally, we split the first term on the RHS of (5.25) into two pieces for some \(\beta \in (0,1)\), and apply the upper-bound estimate (4.25) in Lemma 4.3 to the second piece yieldingThis completes the proof. \(\square \)

We may convert Open image in new window in the last term on the RHS of (5.20) to the error Open image in new window as stated in the following lemma.

Lemma 5.5

Let the problem data satisfy Assumption 1 and Assumption 2. Let \(\varTheta \text { and } K_Z\) the constants given in Proposition 1 and \(C_*\text { and } \hat{C}_*\) the constants given by Lemmas 4.1 and 5.1, respectively. Let
$$\begin{aligned} u&\text { the solution to}~(1.2),&u_1&\text { the solution to}~(2.9), \\ \hat{z}_1&\text { the solution to}~(2.11),&\hat{z}_1^1&\text { the solution to}~(2.10). \end{aligned}$$
Then

Proof

Recall that
$$\begin{aligned} \hat{z}_1&\text { solves } a(\hat{z}_1, v) + \langle b'(u) \hat{z}_1, v\rangle = g(v), ~ \forall v \in {\mathbb {V}}_1, \end{aligned}$$
(5.27)
$$\begin{aligned} \hat{z}_1^1&\text { solves } a(\hat{z}_1^1, v) + \langle b'(u_1) \hat{z}_1^1, v\rangle = g(v), ~ \forall v \in {\mathbb {V}}_1. \end{aligned}$$
(5.28)
Subtracting (5.28) from (5.27) and rearranging terms, we get for any \(v \in {\mathbb {V}}_1\):
$$\begin{aligned} a(\hat{z}_1 - \hat{z}_1^1, v) + \langle (b'(u) - b'(u_1)) \hat{z}_1, v\rangle = \langle b'(u_1)(\hat{z}_1^1 - \hat{z}_1), v\rangle . \end{aligned}$$
(5.29)
In particular, for \(v = \hat{z}_1 - \hat{z}_1^1 \in {\mathbb {V}}_1\) Eq. (5.29) yieldswhere in the last inequality, we used the monotonicity assumption of b in Assumption (1). Now applying the Lipschitz property of \(b'\), the a priori\(L_{\infty }\) bounds on the dual solution \(\hat{z}_1\) (cf. Proposition 1), and both primal and dual \(L_2\) lifting in (5.30), we obtainfrom which the result follows. \(\square \)

Remark 5.2

The result still holds with a modified constant if we weaken the monotonicity assumption as described in Remark 2.1.

Now we are in position to show the contraction of the GOAFEM in terms of the combined quasi-error which is a linear combination of the energy errors and error estimators in primal and limiting dual problems.

Theorem 5.1

(Contraction of GOAFEM) Let the problem data satisfy Assumption 1 and Assumption 2. Let
$$\begin{aligned} u&\text { the solution to}~(1.2),&u_j&\text { the solution to}~(2.9), \\ \hat{z}&\text { the solution to}~(2.6),&\hat{z}_j&\text { the solution to}~(2.11). \end{aligned}$$
Let \(\theta \in (0,1]\), and let \(\{{\mathcal {T}}_j, {\mathbb {V}}_j\}_{j \ge 0}\) be the sequence of meshes and finite element spaces produced by GOAFEM. Let \(\gamma _p > 0\) as given by Theorem 4.2. Then for sufficient small mesh size \(h_{0}\), there exist constants \(\gamma > 0, \pi > 0 \text { and } \alpha _D \in (0, 1)\) such that

Proof

For simplicity, we denote \( \eta _0 = \eta _0(\mathbf{D}, {\mathcal {T}}_0) \) and for \(k = 1\) or 2, denote \(\zeta _{k}(\hat{z}_k) = \zeta _{k}(\hat{z}_k, {\mathcal {T}}_k). \) By the estimator reduction for the limiting dual problem (4.39), for arbitrary \(\delta > 0\) we havewhere \(\lambda = 1- 2^{-1/d}.\) Recall the quasi-orthogonality estimate in the limiting dual problem from Lemma 5.2Adding (5.34) to a positive multiple \(\gamma \) (to be determined) of (5.33) and applying the results of Lemmas 5.4 and 5.5 obtainWe first set \(\gamma \,\,{:=}\,\, (1 + \delta ^{-1})^{-1} \varLambda _1^{-1}\eta _0^{-2}\) to eliminate the last term in  (5.35). This yieldswhere the coefficients \(A \text { and } M\) of (5.36) are given by
$$\begin{aligned} A&= \hat{\varLambda }_G - (1-\beta ) \lambda \theta ^2 \delta (1 + \delta _4)^{-1} C_1^{-2} \varLambda _1^{-1} \eta _0^{-2}, \end{aligned}$$
(5.37)
$$\begin{aligned} M&= (1 + \delta )(1 -\beta \lambda \theta ^2 (1 + \delta _4)^{-1} ), \end{aligned}$$
(5.38)
where \(\delta _{4}\) satisfies \( (1 + \delta _4 ) \,\,{:=}\,\ (1 + \delta _1 )(1 + \delta _2 )(1 + \delta _A )(1 + \delta _B ) \) as was given in (5.24).
For contraction, we require \(A<1\) and \(M<1\) for the coefficients defined by (5.37) and (5.38), that is, we need to choose a \(\beta \in (0, 1)\) such that
$$\begin{aligned} \frac{\delta }{1 + \delta } \frac{1 + \delta _4}{\lambda \theta ^2} < \beta < 1 - \frac{(\hat{\varLambda }_G - 1) \varLambda _C}{\delta } \frac{1 + \delta _4}{\lambda \theta ^2}, \end{aligned}$$
(5.39)
with \(\varLambda _C \,\,{:=}\,\, C_1^2 \varLambda _1 \eta _0^2.\) To demonstrate the existence of such a \(\beta \), set
$$\begin{aligned} \delta _4 = \delta = \frac{1}{2}\lambda \theta ^2. \end{aligned}$$
(5.40)
Then we require the mesh size \(h_{0}\) sufficiently small, such that
$$\begin{aligned} \hat{\varLambda }_G < 1+ \frac{\lambda ^{2}\theta ^{4}}{2(2+\lambda \theta ^{2})\varLambda _{C}}, \end{aligned}$$
(5.41)
for the given \(\theta \in (0,1)\). Note the conditions (5.40) and (5.41) guarantee the interval in (5.39) is nonempty, so there exists a \(\beta \) such that
$$\begin{aligned} \frac{1}{2} < \beta < 1 - \frac{(\hat{\varLambda }_G - 1) \varLambda _C}{\lambda \theta ^2} \left( 1+ \frac{2}{\lambda \theta ^2}\right) . \end{aligned}$$
It remains to control the last term in (5.36). For simplicity, we assume \(\delta _1 = \delta _2 = \delta _A = \delta _B \,{=:}\,\delta _C\). Then the coefficient D in (5.36) is given by
$$\begin{aligned} D&= \delta \lambda \varTheta ^2 K_Z^2 C_*^2 h_0^{2s} \left( \frac{ \theta ^2 +(1+ \delta _C)^2}{(1 + \delta _C)^2 \delta _C} \right) \nonumber \\&\;\;\;\times \left( \frac{h_0^2}{\varLambda _1 \eta _0^2 (1+\delta _{C})} + \hat{C}_*^2 h_0^{2s} \right) . \end{aligned}$$
(5.42)
To control the primal error term with the coefficient D as given by (5.42), we add a positive multiple \(\pi \) (to be determined) of the primal contraction result (4.40) of Theorem 4.2 to (5.37) yieldingWe choose \(\pi \) to ensure \(D + \alpha ^2 \pi < \pi \), namely, \(\pi > \frac{D}{1 - \alpha ^2}\), and set
$$\begin{aligned} \alpha _D^2 \,\,{:=}\,\, \max \left\{ A, M, \frac{D + \alpha ^2 \pi }{\pi }, \alpha ^2 \right\} < 1. \end{aligned}$$
(5.44)
Then the combined quasi-error satisfies the contraction property (5.32). \(\square \)
For simplicity, we denote bythe combined quasi-error in (5.32). The following corollary gives the contraction of the error in the goal function, which is determined by the contraction of the combined quasi-error.

Corollary 5.3

Let the assumptions in Theorem 5.1 hold. Then the error in the goal function is controlled by a constant multiple of the square of the combined quasi-error, i.e.,
$$\begin{aligned} |g(u) - g(u_j)| \le C \bar{Q}^2_j(u_j, \hat{z}_j) \le \alpha _D^{2j} C \bar{Q}_0^2(u_0, \hat{z}_0). \end{aligned}$$
(5.45)

Proof

Choosing the test function \(v = u - u_{j}\) in (2.6), and by linearity and Galerkin orthogonality for the primal problem, we obtain
$$\begin{aligned} g(u) - g(u_j)&= a(\hat{z},u) + \langle b'(u) \hat{z}, u \rangle \nonumber \\&\quad - a(\hat{z}, u_j) - \langle b'(u) \hat{z}, u_j\rangle \nonumber \\&= a(u - u_j, \hat{z}) + \langle b'(u)(u - u_j), \hat{z} \rangle \nonumber \\&= a(u - u_j, \hat{z}) + \langle {\mathcal {B}}_j(u - u_j), \hat{z} \rangle \nonumber \\&\quad + \langle (b'(u) - {\mathcal {B}}_j)(u - u_j), \hat{z} \rangle \nonumber \\&= a(u\! -\! u_j, \hat{z} \!-\! \hat{z}_j) \!+\! \langle b(u) \!-\! b(u_j), \hat{z} \!-\! \hat{z}_j \rangle \nonumber \\&\quad + \langle (b'(u) - {\mathcal {B}}_j)(u - u_j), \hat{z} \rangle . \end{aligned}$$
(5.46)
The third term in the last line of (5.46) represents the error induced by switching from (2.6) to (2.3). This term may be bounded in terms of the constants and \(L_\infty \) estimates in Proposition 1 and
$$\begin{aligned} ||{b'(u) - {\mathcal {B}}_j} ||_{L_2}&= \left\| { \int _0^1 b'(u) - b' \left( u_j + \xi (u - u_j) \right) d\xi }\right\| _{L_2} \\&\le \frac{\varTheta }{2}||{u - u_j} ||_{L_2}, \end{aligned}$$
yielding
$$\begin{aligned}&\langle (b'(u) - {\mathcal {B}}_j)(u - u_j), \hat{z} \rangle \nonumber \\&\quad \le K_Z ||{b'(u) - {\mathcal {B}}_j} ||_{L_2}||{u - u_j} ||_{L_2} \nonumber \\&\quad \le \frac{1}{2} \varTheta K_Z ||{u - u_j} ||_{L_2}^2. \end{aligned}$$
(5.47)
Then by (5.46), (5.47), the Cauchy–Schwarz inequality and \(L_2\)-lifting as in Lemmas 4.1 and 5.1Therefore the error in the goal function is bounded above by a constant multiple of the square of the combined quasi-error \(\bar{Q}^2(u_j, \hat{z}_j)\). Thus (5.45) follows by the contraction result in Theorem 5.1. \(\square \)

6 Numerical experiments

In this section we present some numerical experiments implemented using FETK [21], which is a fairly standard set of finite element modeling libraries for approximating the solutions to systems of nonlinear elliptic and parabolic equations. We compare three methods: HPZ, the algorithm presented in this paper; MS, the algorithm presented in [34]; and the DWR, the dual weighted residual method as described in, for example [2, 5, 14, 15, 19, 20]. We see HPZ performs with comparable efficiency to MS, with the added benefit of fewer iterations of the adaptive algorithm (3.1) resulting in a shorter overall runtime. The efficiency of the residual based algorithms HPZ and MS in comparison to DWR varies with the problem structure. The examples below show cases where each algorithm may outperform the others, but where the performance of all three is comparable with a small change in the problem parameters. In the figures below we show efficiency by comparing the error to the number of elements in the adaptive mesh; in the corresponding data tables, we also include mesh degrees of freedom which is correspondingly lower for the residual based methods than it is for DWR. This last measure is of increasing importance for nonlinear problems for which the majority of practical runtime is spent solving Newton iterations of linear systems.

In our DWR implementation, the finite element space \({\mathbb {V}}_{k}\) for the primal problem employs linear Lagrange elements as do HPZ and MS for both the primal and dual spaces. For DWR, the dual finite element space \({\mathbb {V}}_{k}^2\) uses quadratic Lagrange elements. For any \(v \in {\mathbb {V}}_{k}\), the elementwise DWR indicator defined as
$$\begin{aligned} \eta _{k}^D (v,T) \,\,{:=}\,\, \langle R(v), z^2 - I_k z^2\rangle _T + \frac{1}{2} \langle J_T(v), z^2 - I_k z^2 \rangle _{\partial T}. \end{aligned}$$
The DWR indicator estimates the influence of the dual solution on the primal residual. Here \(z^2 \in {\mathbb {V}}_{k}^2\) is the solution of the approximate dual problem (2.10) and \(I_k\) is the interpolator onto \({\mathbb {V}}_{k}\). Then the DWR error estimator is the absolute value of the sum of indicators
$$\begin{aligned} \eta _k^D = \left| \sum _{T \in {\mathcal {T}}_k} \eta _T^D(u_k,T) \right| \le \sum _{T \in {\mathcal {T}}_k} \left| \eta _T^D(u_k,T)\right| . \end{aligned}$$
Both HPZ and MS use the residual based indicators (3.3) and (3.4) for primal and approximate dual problems, respectively.

In the adaptive algorithms, we use the Dörfler marking strategy (3.8)–(3.9) with parameter \(\theta = 0.6.\) For the nonlinear primal problem, at each refinement we use a Newton-type iteration to solve the resulting nonlinear system of algebraic equations, which reduces the nonlinear residual to the tolerance \( \Vert F(u)\Vert _{L_{2}} \le 10^{-7}\). On the initial triangulation, we use a zero initial guess for the Newton iteration; then for each subsequent refinement, we interpolate the numerical solution from the previous step to the current triangulation and then use it as the initial guess for the Newton iteration. By doing this, we have a good initial guess for the Newton iteration indicating a quadratic convergence rate of the nonlinear iterations.

In the following examples, we use the same primal problem given in weak form by
$$\begin{aligned} \frac{1}{1000}\langle \nabla u, \nabla v \rangle + \langle 3 u^3, v \rangle = f(v). \end{aligned}$$
The problem data for each problem are defined by \(g(u) = \int _\varOmega g u\) and \(f(v) = \int _\varOmega f v\) where \(g= g(x,y)\), \(f=f(x,y)\) are defined in each example over the domain \(\varOmega = (0,1)^2\). The initial triangulation is a uniform mesh consisting of 144 elements. Here we consider problems where the primal and dual data and likewise the primal and dual solutions contain either sharp spikes or shallower bumps where these functions feature rapidly changing gradients.

Example 1

(Separated primal data) This problem features a single Gaussian spike as the goal function g(xy) and primal data focused on two bumps, one of which overlaps with the spike in g(xy). We look at two sets of parameters featuring different placement of the second primal bump. This problem demonstrates the difference between the algorithms when some or all of the primal data has a strong influence on the quantity of interest g(u) and is remote from the spike in the dual solution.

The goal function is given by \(g(x,y) = a \exp (-a [(x - x_d)^2 + (y - y_d)^2])\), with \(a = 200\). The primal data f(xy) is chosen so the exact solution u is
$$\begin{aligned} u(x,y) =&\sin (2\pi x)\sin (2\pi y) \\&\times \left\{ \frac{1}{2[(x-x_0)^2+(y-y_0)^2] + 10^{-2} } \right. \\&+\left. \frac{1}{2[(x-x_1)^2+(y-y_1)^2] + 10^{-2} } \right\} . \end{aligned}$$
The fixed parameters \((x_0,y_0) = (0.7, 0.7)\) and \((x_d, y_d) = (0.7, 0.7)\) determine an interaction between the primal solution and the dual data, and the second primal spike \((x_1,y_1)\) is tested at a near and far location.
$$\begin{aligned} (x_1,y_1) = (0.6,0.6) , \end{aligned}$$
(6.1)
$$\begin{aligned} (x_1,y_1) = (0.3,0.3) . \end{aligned}$$
(6.2)
Figure 1 shows the performance of the three algorithms with the parameters of (6.1) and (6.2). The performance of all algorithms is better than \(n^{-1}\), with n the number of mesh elements, and we also see that with the partial overlap of primal and dual data as in (6.1) the residual-based methods produce better efficiency as the locations of large error in both primal and dual residuals are good indicators of error in the goal function. Figure 2 shows the residual based methods refine for both bumps in the primal data whereas DWR refines only where the the primal and dual data largely coincide; the partitions are qualitatively distinct while the error reduction in the goal function is similar in all three methods.
Fig. 1

Top: goal error after 19 HPZ, 33 MS and 19 DWR iterations for  (6.1), compared with \(n^{-1}\). Bottom: goal error after 18 HPZ, 36 MS and 19 DWR iterations for parameter set (6.2), compared with \(n^{-1}\) with n the number of elements

Fig. 2

Top: 9 iterations of HPZ (726 elements). Center: 17 iterations of MS (678) elements). Bottom: 10 iterations of DWR (600 elements) for parameter set (6.2)

For practical use, the dual indicator may be used to predict the error: the results for the problem (6.2) are summarized in Table 1 for HPZ, DWR and MS, where we consider the iterations necessary to drive the dual indicator \(\eta _d\) below a given tolerance. For HPZ and MS, \(\eta _d\) is the residual based estimator for the dual problem and for DWR, \(\eta _d\) is the DWR estimator. The primal degrees of freedom, DOF (P); and dual degrees of freedom, DOF (D) are equal in HPZ and MS which use linear basis functions for both problems but DOF (D) is higher for DWR which uses quadratic basis functions for the dual problem.

In Table 1 we see comparable error reduction in the three methods given the reduction of the error estimator, with MS requiring nearly twice as many adaptive iterations as HPZ and in this case fewer degrees of freedom. Significantly, the DWR method in this example requires approximately twice as many primal degrees of freedom as HPZ, and more than eight times as many dual degrees of freedom to drive the error estimator beneath the same tolerance.
Table 1

Comparison of HPZ, DWR and MS for (6.2) with respect to driving the dual estimator \(\eta _d\) below a given tolerance

 

Tol

\(10^{-1}\)

\(10^{-2}\)

\(10^{-3}\)

\(10^{-4}\)

Iterations

HPZ

5

8

13

18

DWR

9

12

17

21

MS

6

14

24

34

DOF (P)

HPZ

139

286

1183

5437

DWR

235

583

3111

11,881

MS

111

232

887

4184

DOF (D)

HPZ

139

286

1183

5437

DWR

913

2305

12,415

47,493

MS

111

232

887

4184

\(\eta _{d}\)

HPZ

4.2778e\(-\)02

8.1993e\(-\)03

7.2718e\(-\)04

9.0042e\(-\)05

DWR

6.3575e\(-\)02

9.1835e\(-\)03

5.4107e\(-\)04

7.3183e\(-\)05

MS

8.8195e\(-\)02

8.2971e\(-\)03

7.1939e\(-\)04

8.8848e\(-\)05

\(|g(u) - g(u_{k})|\)

HPZ

1.3422e\(-\)04

3.6785e\(-\)05

1.5103e\(-\)06

9.7768e\(-\)08

DWR

1.0092e\(-\)04

1.8723e\(-\)05

6.7427e\(-\)07

1.4926e\(-\)08

MS

3.1049e\(-\)04

6.5948e\(-\)05

2.5896e\(-\)06

1.5510e\(-\)07

Example 2

(Goal function with two spikes) In this example we consider the problem with a single spike in the primal data and a goal function g(xy) consisting of a Gaussian average about two separated points. We keep the far point fixed and move the second point closer to the spike in the primal data to investigate which of the algorithms are more effective as we introduce and overlap of the refinement sets based on the primal and dual problems. In contrast to parameter sets (6.1)–(6.2), DWR fares as well or better than the residual based methods for (6.3)–(6.4).

We observe that varying relative distribution of data in the primal and dual problems changes the relative efficiency of the three algorithms.The goal function is given by
$$\begin{aligned} g(x,y)= & {} a \exp (-a((x-x_0)^2 + (y - y_0)^2))\\&+ a\exp (-a((x-x_1)^2 + (y - y_1)^2)), \end{aligned}$$
with \(a = 400\) and \((x_0, y_0) = (0.7, 0.7)\). The data f(xy) is chosen so the exact solution u(xy) is given by
$$\begin{aligned} u = \sin (\pi x) \sin ( \pi y) \frac{1}{ 2[(x - x_p)^2 + (y - y_p)^2] + 10^{-3}} \end{aligned}$$
with \((x_p,y_p) = (0.3, 0.3)\) and the following locations of \((x_1,y_1):\)
$$\begin{aligned} (x_1,y_1)&= (0.7,0.3) ,\end{aligned}$$
(6.3)
$$\begin{aligned} (x_1,y_1)&= (0.55,0.3) . \end{aligned}$$
(6.4)
For the parameter set (6.3) shown in Fig. 3 both dual spikes are remote from the primal data. As seen in Fig. 4 each algorithm displays a distinct trend in its adaptive refinement: HPZ refines for both primal and dual; MS refines for both with a bias towards the primal, with 17 primal refinements and 10 dual refinements in this case; DWR refines with a bias towards the dual data. HPZ and DWR show similar goal error reduction while MS stalls at least on these relatively early refinements by neglecting to refine for the dual problem. In parameter set (6.4) DWR displays the better error reduction based on the number of elements in the mesh. To investigate the behavior of the estimator in the asymptotic regime for (6.3) we look at number of iterations, total degrees of freedom used and the error in the goal function based on driving the scaled indicator below a given tolerance. Here, \(\eta _d\) is the residual based dual indicator for HPZ and the DWR indicator for DWR; we scale \(\eta _d\) by \(\eta _{d,10}\), the value of \(\eta _d\) at the tenth iteration, and consider the tolerance in finer increments to better describe the behavior.
Fig. 3

Top: goal error after 19 HPZ, 46 MS and 19 DWR iterations for  (6.3). Bottom: goal error after 19 HPZ, 44 MS and 19 DWR iterations for  (6.4), compared with \(n^{-1}\), where n is the number of elements

Fig. 4

Top: 13 iterations of HPZ (1187 elements). Center: 26 iterations of MS (1156) elements). Bottom: 14 iterations of DWR (1115 elements) for  (6.3)

In this case, the dof for HPZ are intermediate between the primal and dual dof for DWR, and we see the indicators and error decrease at a similar rate. The MS method was not included in Table 2 as the estimator did not achieve a steady asymptotic decrease: consistent with the theory, the MS dual indicator may increase on iterations where the algorithm refines for the primal marked set. For instance, Table 3 shows a small subset of data in which we see the dual indicator increasing but the error decreasing at iterations 15 and 17, and the dual indicator decreasing with the error increasing at iteration 16, and both decreasing at iteration 18.
Table 2

Comparison of HPZ and DWR for (6.3) with respect to driving the scaled dual estimator \(\eta _d/\eta _{d,10}\) below a given tolerance, where \(\eta _{d,10}\) is the dual estimator at the tenth adaptive iteration

 

Tol

\(2^{0}\)

\(2^{-1}\)

\(2^{-2}\)

\(2^{-3}\)

Iterations

HPZ

10

13

16

20

DWR

10

12

15

18

DOF (P)

HPZ

426

1187

3770

19,482

DWR

429

789

2407

8325

DOF (D)

HPZ

426

1187

3770

19,482

DWR

1689

3129

9601

33,269

\(\eta _{d}/\eta _{d,10}\)

HPZ

1.0000e\(+\)00

4.7192e\(-\)01

2.4389e\(-\)01

1.0318e\(-\)01

DWR

1.0000e\(+\)00

4.6392e\(-\)01

2.0621e\(-\)01

1.0384e\(-\)01

\(|g(u) - g(u_{k})|\)

HPZ

1.5260e\(-\)05

3.7866e\(-\)05

4.8382e\(-\)06

6.8652e\(-\)07

DWR

3.8193e\(-\)05

6.6045e\(-\)06

4.8778e\(-\)06

2.2367e\(-\)07

Table 3

Nonmonotonic behavior of dual error estimator and goal error in MS for (6.3)

ITER

DOF

\(\eta _d\)

\(|g(u) - g(u_k)|\)

14

150

1.0277e\(+\)00

2.2820e\(-\)02

15

159

1.1230e\(+\)00

7.1823e\(-\)03

16

172

6.9759e\(-\)01

8.2955e\(-\)03

17

191

7.0258e\(-\)01

7.6307e\(-\)04

18

204

3.6112e\(-\)01

3.1665e\(-\)04

These examples indicate that while the norm of the residual does overestimate the error in the quantity of interest, refining the mesh with respect to the residual based indicators is an effective method to reduce the error in the goal function as long as both primal and dual indicators are taken into account at each iteration: in particular, the norm-based indicators effectively locate the elements with large error in the goal function yielding an efficient goal-oriented algorithm. The last example also illustrates the limitation of the efficiency of the MS algorithm which was designed for linear symmetric problems and does not necessarily extend to more general and in this case nonlinear problems. Here the method stalls because the consistently smaller refinement sets correspond to the primal residual, neglecting the error in the dual problem.

The effectiveness of the DWR method is based on the assumption that \(\langle R(u_h),\tilde{z} -z_h \rangle \) is a good predictor for the error \(g(e_h)\) for \(\tilde{z}\) an approximation to z not in the primal finite element space. This appears to work so long as rapidly changing gradients in the dual solution coincide spatially with spikes in g(xy), and the primal residual \(R(u_h)\) captures sufficient information about the primal solution in the vicinity of the influence function. For an example of where the first condition fails, we refer to the linear convection-diffusion problem discussed in [24], and a demonstration of the second condition is (6.1). Under certain conditions, namely a confined region where the spikes in primal data and dual solution overlap that coincides with the overlap in the spikes in the primal solution and dual data, the DWR methods outperforms the residual based methods.

In many cases, all three goal-oriented methods display similar performance, yet with qualitatively different adaptive partitions. The relative performances of HPZ and MS do appear to be dependent on the structure of the primal problem, however it is not clear at this stage how to predict which algorithm will yield a better reduction in goal error. In problems where the HPZ and MS results appear similar, we note that MS takes considerably longer to run and may require nearly twice as many total iterations of the algorithm which is problematic as most of the runtime each iteration is spent on nonlinear solves. This same issue can increase the runtime of the DWR algorithm as the dual problem assembled with quadratic basis functions is approximately four times the size of the primal.

7 Conclusion

In this article we developed convergence theory for a class of goal-oriented adaptive finite element algorithms for second order semilinear elliptic equations. We first introduced several approximate dual problems, and briefly discussed the target problem class. We then reviewed some standard facts concerning conforming finite element discretization and error-estimate-driven adaptive finite element methods (AFEM). We included a brief summary of a priori estimates for semilinear problems, and described goal-oriented variations of the standard approach to AFEM (GOAFEM). Following the recent work of Mommer-Stevenson and Holst-Pollock for linear problems, we established contraction of GOAFEM for the primal problem. We developed additional estimates that make it possible to establish contraction of the combined quasi-error, and showed convergence in the sense of the quantity of interest. Some simple numerical experiments confirmed these theoretical predictions and demonstrated that our method performs comparably to other standard adaptive goal-oriented strategies, and has the additional advantage of provable convergence for problems where the theory has not been developed for the other two methods. Our analysis was based on the recent contraction frameworks for the semilinear problem developed by Holst, Tsogtgerel, and Zhu and Bank, Holst, Szypowski and Zhu and those for linear problems as in Cascon, Kreuzer, Nochetto and Siebert, and Nochetto, Siebert, and Veeser. In addressing the goal-oriented problem we considered the approaches of Mommer and Stevenson for symmetric linear problems and Holst and Pollock for nonsymmetric problems. Unlike the linear case, we were faced with tracking linearized and approximate dual sequences in order to establish contraction with respect to the quantity of interest.

In our numerical results we demonstrated the efficiency of the method presented here as compared with the dual-weighted residual method, the method introduced by Mömmer and Stevenson in [34] and a standard residual-based adaptive finite element method. We emphasize the goal of an adaptive method is not to refine the fewest elements possible per iteration but rather to optimize the efficiency of the method as measured by degrees of freedom necessary to attain a given accuracy. This efficiency can only be optimized by an adaptive method that responds to all dominant sources of error at each iteration; while problems can be designed for which each of the presently considered methods can miss at least one dominant source of error in the mesh, the method presented in this paper compares well to the others in overall efficiency, and has the added benefit of analytical convergence.

In the present paper we assume the primal and approximate dual solutions are solved on the same mesh at each iteration. The determination of strong convergence results for a method which solves the primal (nonlinear) problem on a coarse mesh and the dual on a fine mesh is the subject of future investigation.

Notes

Acknowledgments

MH was supported in part by NSF Awards 1065972, 1217175, 1262982, 1318480, and by AFOSR Award FA9550-12-1-0046. SP and YZ were supported in part by NSF Awards 1065972 and 1217175. YZ was also supported in part by NSF DMS 1319110, and in part by University Research Committee Grant No. F119 at Idaho State University, Pocatello, Idaho.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California at San DiegoSan DiegoUSA
  2. 2.Department of MathematicsTexas A&M UniversityCollege StationUSA
  3. 3.Department of MathematicsIdaho State UniversityPocatelloUSA

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