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

- 148 Downloads
- 3 Citations

## 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

*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.

- 1.Quasi-orthogonality: There exists \( \varLambda _G > 1\) such that
- 2.Error estimator as upper bound on error: There exists \(C_1 > 0\) such that
- 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 \(\delta > 0\)$$\begin{aligned} \eta _2^2(v_2,{\mathcal {T}}_2) \le&(1 + \delta ) \{ \eta _1^2(v_1, {\mathcal {T}}_1) - \lambda \eta _1^2(v_1, {\mathcal {M}}) \} \\&+(1 + \delta ^{-1}) \varLambda _1 \eta _0^2 | \! | \! | {v_2 - v_1} | \! | \! |^2. \end{aligned}$$

*combined quasi-error*which 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.

**Assumption 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)\(b: \varOmega \times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is smooth on the second argument. Here and in the remainder of the paper, we write
*b*(*u*) instead of*b*(*x*,*u*) 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)
\(f\in L_2(\varOmega )\).

*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 \(-\nabla \cdot (A\nabla )\).

### 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.

*u*, the residual is given by

*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

*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

*T*. For any subset \({\mathcal {S}}\subseteq {\mathcal {T}}\),

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**

*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**

- 1)
*b*is Lipschitz on \([u_-,u_+]\cap H_0^1(\varOmega )\) for a.e. \(x \in \varOmega \) with constant*B*. - 2)
\(b'\) is Lipschitz on \([u_-,u_+]\cap H_0^1(\varOmega )\) for a.e. \(x \in \varOmega \) with constant \(\varTheta \).

- 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

* 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.

*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*

**Procedure ESTIMATE.***jump residual*for both the primal and linearized dual problems is:

*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

*The Dörfler marking strategy for the goal-oriented problem is based on the following steps as in [34]:*

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

* 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**

*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

*Proof*

*b*(Proposition 1):

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

**Corollary 4.1**

*Proof*

*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

*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**

*Proof*

*b*(cf. Proposition 1).

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**

### 4.3 Estimator reduction

*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

*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]):

**Lemma 4.4**

*d*and the initial mesh \({\mathcal {T}}_0\).

*Proof*

*e*. The second term in (4.32) is bounded by

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**

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**

*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

**Lemma 5.1**

*y*to (5.1) belongs to \(H^{1+s}(\varOmega ) \cap H_0^1(\varOmega )\) for some \(0 < s \le 1\) such that

*Proof*

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

*Remark 5.1*

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

**Lemma 5.2**

*Proof*

### 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**

*d*and the regularity of the initial mesh \({\mathcal {T}}_0\).

*Proof*

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**

*Proof*

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**

*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

**Lemma 5.4**

*Proof*

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**

*Proof*

*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 obtain

*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**

*Proof*

*D*in (5.36) is given by

*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) yielding

**Corollary 5.3**

*Proof*

## 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 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.

*Example 1*

*(Separated primal data)* This problem features a single Gaussian spike as the goal function *g*(*x*, *y*) and primal data focused on two bumps, one of which overlaps with the spike in *g*(*x*, *y*). 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.

*f*(

*x*,

*y*) is chosen so the exact solution

*u*is

*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.

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.

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*(*x*, *y*) 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).

*f*(

*x*,

*y*) is chosen so the exact solution

*u*(

*x*,

*y*) is given by

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 |

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*(*x*, *y*), 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.

### References

- 1.Axelsson, O., Barker, V.A.: Finite Element Solution of Boundary Value Problems: Theory and Computation. Society for Industrial and Applied Mathematics, Philadelphia, PA (2001)CrossRefGoogle Scholar
- 2.Bangerth, W., Rannacher, R.: Adaptive Finite Element Methods for Differential Equations. Birkhauser, Boston (2003)CrossRefGoogle Scholar
- 3.Bank, R., Holst, M., Szypowski, R., Zhu, Y.: Finite Element Error Estimates for Critical Growth Semilinear Problems Without Angle Conditions (2011) Available as arXiv:1108.3661 [math.NA]. Submitted
- 4.Becker, R., Rannacher, R.: A feed-back approach to error control in finite element methods: basic analysis and examples. East-West J. Numer. Math.
**4**, 237–264 (1996)MathSciNetGoogle Scholar - 5.Becker, R., Rannacher, R.: Weighted a posteriori error control in FE methods. Preprint 96-1, SFB 359, Universitat, pp. 18–22 (1996)Google Scholar
- 6.Becker, R., Rannacher, R.: An optimal control approach to a posteriori error estimation in finite element methods. In: Iserles, A. (ed.) Acta Numer, pp. 1–0102. Cambridge University Press, Cambridge (2001)Google Scholar
- 7.Binev, P., Dahmen, W., DeVore, R.: Adaptive finite element methods with convergence rates. Numer. Math.
**97**(2), 219–268 (2004)MathSciNetCrossRefGoogle Scholar - 8.Brenner, S., Scott, L.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, Berlin (2008)CrossRefGoogle Scholar
- 9.Cascon, J.M., Kreuzer, C., Nochetto, R.H., Siebert, K.G.: Quasi-optimal convergence rate for an adaptive finite element method. SIAM J. Numer. Anal.
**46**(5), 2524–2550 (2008)MathSciNetCrossRefGoogle Scholar - 10.Ciarlet, P.G.: Finite Element Method for Elliptic Problems. Society for Industrial and Applied Mathematics, Philadelphia, PA (2002)CrossRefGoogle Scholar
- 11.Dahmen, W., Kunoth, A., Vorloeper, J.: Convergence of Adaptive Wavelet Methods for Goal-oriented Error Estimation. Sonderforschungsbereich 611, Singuläre Phänomene und Skalierung in Mathematischen Modellen. SFB 611 (2006)Google Scholar
- 12.Dörfler, W.: A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal.
**33**, 1106–1124 (1996)MathSciNetCrossRefGoogle Scholar - 13.Eriksson, K., Estep, D., Hansbo, P., Johnson, C.: Introduction to adaptive methods for differential equations. In: Iserles, A. (ed.) Acta Numer, pp. 105–158. Cambridge University Press, Cambridge (1995)Google Scholar
- 14.Estep, D., Holst, M., Larson, M.: Generalized green’s functions and the effective domain of influence. SIAM J. Sci. Comput.
**26**, 1314–1339 (2002)MathSciNetCrossRefGoogle Scholar - 15.Estep, D., Holst, M., Mikulencak, D.: Accounting for stability: a posteriori error estimates based on residuals and variational analysis. In: Communications in Numerical Methods in Engineering, pp. 200–202 (2001)Google Scholar
- 16.Estep, D., Larson, M.G., Williams, R.D.: Estimating the error of numerical solutions of systems of reaction-diffusion equations. Mem. Am. Math. Soc.
**146**(696), 101–109 (2000)MathSciNetGoogle Scholar - 17.Evans, L.C.: Partial Differential Equations (Graduate Studies in Mathematics, V. 19) GSM/19. American Mathematical Society, New York (1998)Google Scholar
- 18.Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1977)CrossRefGoogle Scholar
- 19.Giles, M., Süli, E.: Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality. Acta Numer.
**11**, 145–236 (2003)Google Scholar - 20.Grätsch, T., Bathe, K.-J.: A posteriori error estimation techniques in practical finite element analysis. Comput. Struct.
**83**(4–5), 235–265 (2005)CrossRefGoogle Scholar - 21.Holst, M.: Adaptive numerical treatment of elliptic systems on manifolds. Adv. Comput. Math.
**15**(1–4), 139–191 (2001) Available as arXiv:1001.1367 [math.NA] - 22.Holst, M.: Applications of domain decomposition and partition of unity methods in physics and geometry. In: Herrera, I., Keyes, D., Widlund, O., Yates, R. (eds.) Proceedings of the Fourteenth International Conference on Domain Decomposition Methods, pp. 63–78. National Autonomous University of Mexico (UNAM) (2003) Available arXiv:1001.1364 [math.NA]
- 23.Holst, M., McCammon, J., Yu, Z., Zhou, Y., Zhu, Y.: Adaptive finite element modeling techniques for the Poisson-Boltzmann equation. Commun. Comput. Phys.
**11**(1), 179–214 (2012) Available arXiv:1009.6034 [math.NA] - 24.Holst, M., Pollock, S.: Convergence of goal oriented methods for nonsymmetric problems . Numer Methods Partial Differ. Equ. (2015) Available as arXiv:1108.3660 [math.NA]
- 25.Holst, M., Szypowski, R., Zhu, Y.: Adaptive finite element methods with inexact solvers for the nonlinear poisson-boltzmann equation. In: Bank, R., Holst, M., Widlund, O., Xu, J. (eds.) Domain Decomposition Methods in Science and Engineering XX, volume 91 of Lecture Notes in Computational Science and Engineering, pp. 167–174. Springer, Berlin (2013)Google Scholar
- 26.Holst, M., Szypowski, R., Zhu, Y.: Two-grid methods for semilinear interface problems. Numer Methods Partial Differ. Equ.
**29**(5), 1729–1748 (2013)MathSciNetCrossRefGoogle Scholar - 27.Holst, M., Tsogtgerel, G., Zhu, Y.: Local and global convergence of adaptive methods for nonlinear partial differential equations (2008) Available as arXiv:1001.1382 [math.NA]
- 28.Jüngel, A., Unterreiter, A.: Discrete minimum and maximum principles for finite element approximations of non-monotone elliptic equations. Numer. Math.
**99**(3), 485–508 (2005)MathSciNetCrossRefGoogle Scholar - 29.Karatson, J., Korotov, S.: Discrete maximum principles for finite element solutions of nonlinear elliptic problems with mixed boundary conditions. Numer. Math.
**99**, 669–698 (2005)MathSciNetCrossRefGoogle Scholar - 30.Kerkhoven, T., Jerome, J.W.: \(L_{\infty }\) stability of finite element approximations of elliptic gradient equations. Numer. Math.
**57**, 561–575 (1990)MathSciNetCrossRefGoogle Scholar - 31.Kesavan, S.: Topics in Functional Analysis and Applications. Wiley, New York, NY (1989)Google Scholar
- 32.Korotov, S.: A posteriori error estimation of goal-oriented quantities for elliptic type bvps. J. Comput. Appl. Math.
**191**(2), 216–227 (2006)MathSciNetCrossRefGoogle Scholar - 33.Mekchay, K., Nochetto, R.: Convergence of adaptive finite element methods for general second order linear elliptic PDE. SINUM
**43**(5), 1803–1827 (2005)MathSciNetCrossRefGoogle Scholar - 34.Mommer, M.S., Stevenson, R.: A goal-oriented adaptive finite element method with convergence rates. SIAM J. Numer. Anal.
**47**(2), 861–886 (2009)MathSciNetCrossRefGoogle Scholar - 35.Moon, K.-S., von Schwerin, E., Szepessy, A., Tempone, R.: Convergence rates for an adaptive dual weighted residual finite element algorithm. BIT
**46**(2), 367–407 (2006)MathSciNetCrossRefGoogle Scholar - 36.Nochetto, R.H., Siebert, K.G., Veeser, A.: Theory of Adaptive Finite Element Methods: An Introduction. Springer, Berlin (2009)Google Scholar
- 37.Oden, J., Prudhomme, S.: Goal-oriented error estimation and adaptivity for the finite element method. Comput. Math. Appl.
**41**, 735–756 (2001)MathSciNetCrossRefGoogle Scholar - 38.Sewell, E.G.: Automatic generation of triangulations for piecewise polynomial approximation. Ph. D. dissertation. Purdue Univ., West Lafayette, IN (1972)Google Scholar
- 39.Strang, G., Fix, G.J.: An Analysis of the Finite Element Method. Prentice-Hall (Series in Automatic Computation), Englewood Cliffs, NJ (1973)Google Scholar
- 40.Struwe, M.: Variational Methods, 3rd edn. Springer, Berlin (2000)CrossRefGoogle Scholar
- 41.Verfürth, R.: A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations. Math. Comput.
**62**(206), 445–475 (1994)CrossRefGoogle Scholar - 42.Verfürth, R.: A review of a posteriori error estimation and adaptive mesh refinement tecniques. Teubner–Wiley, Stuttgart (1996)Google Scholar