Abstract
In this paper we introduce and analyze the residual-based a posteriori error estimation of the partially penalized immersed finite element method for solving elliptic interface problems. The immersed finite element method can be naturally utilized on interface-unfitted meshes. Our a posteriori error estimate is proved to be both reliable and efficient with both reliability and efficiency constants independent of the location of the interface. Numerical results indicate that the error estimation is robust with respect to the coefficient contrast.
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1 Introduction
Interface problems arise widely in the multi-physics and multi-material applications in fluid mechanics and material science. The governing partial differential equations (PDEs) for interface problems are usually characterized with discontinuous coefficients that represent different material properties. The solutions to the interface problems often involve kinks, singularities, discontinuities, and other non-smooth behaviors. It is therefore challenging to obtain accurate numerical approximations for interface problems. Moreover, the complexity of the interface geometry may add an extra layer of difficulty to the numerical approximation.
In general, there are two classes of numerical methods for solving interface problems. The first class of methods use interface-fitted meshes, i.e., the meshes are tailored to fit the interface, see the left plot in Fig. 1. Methods of this type include classical finite element methods [14], discontinuous Galerkin methods [4] and recently developed weak Galerkin methods [30], to name a few. The second class of methods use unfitted meshes which are independent of the interface, as illustrated in the right plot in Fig. 1. In the past few decades, many numerical methods based on unfitted meshes have been developed. In the finite difference framework, since the pioneering work of immersed boundary method [31] by Peskin, many numerical methods of finite difference type have been developed such as immersed interface method [20, 23], matched interface and boundary method [35]. In the finite element framework, there are quite a few numerical methods developed, for instance, the general finite element method [5], unfitted finite element method [17], multi-scale finite element method [19], extended finite element method [29], and immersed finite element method (IFEM) [22, 24]. A great advantage for unfitted numerical methods is that they can circumvent (re)meshing procedure which can be very expensive especially for time-dependent problems with complex interface geometry or for shape optimization processes that require repeated updates of the mesh.
An interface-fitted mesh (left), and an unfitted mesh (right)
The IFEM was first developed in [22] for a one-dimensional elliptic interface problem and then extended to higher-order approximations [1, 2, 10, 11] and to higher-dimensional elliptic interface problems [16, 18, 21, 26, 32, 34]. Recently, the partially penalized immersed finite element method was introduced in [25]. Compared to classical IFEM, the partially penalized IFEM contains normal flux jump terms on interface edges to ensure the consistency of the scheme. In addition, the new IFEM includes a stabilization term on interface edges to guarantee the stability of the scheme. The partially penalized IFEM significantly improves the numerical approximation, especially the accuracy around the interface. The optimal a priori error estimate is theoretically proved for partially penalized IFEM in the energy norm [25].
We note that the partially penalized IFEM can solve elliptic interface problems accurately on uniform Cartesian meshes provided that the exact solution of interface problems is piecewise smooth, and the contrast of the coefficient is “moderate”. However, for interface problems that also involve singularity or steep gradient, the partially penalized IFEM alone may not be efficient to obtain an accurate approximation on uniform meshes. In such cases, it is necessary to apply certain adaptive mesh refinement (AMR) strategy to IFEM. The goal for AMR is to obtain an approximate solution within prescribed error tolerance with the minimum computational cost which is particularly rewarding for interface problems with non-smooth solutions.
The key success for AMR is the a posteriori error estimation which provides both global and local information on the approximation error. Moreover, in many applications even without the intention of performing adaptive mesh refinement, the a posteriori error estimation is also important in assessing the quality of the simulation by providing an effective error control. We note that the a priori error estimation of IFEM is getting mature in the past decade, but the a posteriori error estimation of IFEM is still in the infancy. In this paper, we will develop and analyze a residual-based a posteriori error estimate for the partially penalized IFEM for second-order elliptic interface problems in the two-dimensional space.
Comparing to the residual-based error estimation for the classical finite element solution [6], the newly introduced estimator in this work additionally includes the jump of the tangential derivative of numerical solutions on interface edges besides the standard terms of element residual and jump of normal flux on all edges. This is necessary since the numerical solution is in general discontinuous across the interface edges due to the construction of the IFEM basis functions. Moreover, the new error estimator also includes the geometrical fitting error due to the polygonal approximation of the curved interface. Theoretically, we prove that the error estimation is both globally reliable and locally efficient. To prove reliability, we use the Helmholtz decomposition and an \(L^2\) representation technique recently introduced in [8, 9]; moreover, we introduce a new type of Clément-type interpolation in the IFEM space that allows us to take advantage of the error equation of IFEM. In the efficiency analysis for IFEM, the technique using the standard bubble functions [33] is invalid because the edge jumps of the normal flux and the tangential derivative become piecewise constant on the interface edges. Instead, we prove the efficiency in two different approaches that aim to provide an optimal efficiency constant for both regular and irregular interface edges.
The rest of the article is organized as follows. In Sect. 2, we recall the partially penalized IFEM for elliptic interface problems. In Sect. 3, we introduce our residual-based error estimator specially designed for IFEM. Sections 4 and 5 are dedicated to the analysis of global reliability and local efficiency, respectively. Finally, in Sect. 6, we present several numerical experiments to test the performance of our a posteriori error estimators.
2 Interface Problems and Partially Penalized IFEM
Let \(\Omega \subset \mathbb {R}^2 \) be a polygonal domain with Lipschitz boundary \(\partial \Omega = {\overline{\Gamma }}_D \cup {\overline{\Gamma }}_N\), where \( \Gamma _D \cap \Gamma _N = \emptyset \). Assume that \(\text{ meas }(\Gamma _D) > 0\). We consider the elliptic interface problem:
with boundary conditions
Here, \(f \in L^2(\Omega )\), \(g_{_N}\in L^2(\Gamma _N)\), and \(\mathbf{n}\) is the unit vector outward normal to \(\partial \Omega \). The notations \(\nabla \) and \(\nabla \cdot \) are the gradient and divergence operators, respectively. Furthermore, assume that \(\Omega \) is separated by a closed smooth interface curve \(\Gamma \) into \(\Omega ^+\) and \(\Omega ^-\) such that \(\overline{\Omega } = \overline{\Omega ^+\cup \Gamma \cup \Omega ^-}\). The diffusion coefficient \(\alpha \) is assumed to be a positive piecewise constant function as follows
Denote by \(\rho = \dfrac{\alpha ^+}{\alpha ^-}\) the ratio of the coefficient jump. The solution is assumed to satisfy the following interface jump conditions:
where the jump of a function v across the interface \(\Gamma \) is defined by
We use the standard notations for the Sobolev spaces. Let
Then the variational problem for (2.1) is to find \(u \in H_D^1(\Omega )\) such that
where \((\cdot ,\cdot )_\omega \) is the \(L^2\) inner product on \(\omega \). The subscript \(\omega \) is omitted when \(\omega =\Omega \).
2.1 Triangulation
In this paper, we only consider the triangular meshes in two dimensions. Let \({\mathcal T}=\{K\}\) be a triangulation of \(\Omega \) that is regular but not necessarily body-fitted. Denote the set of all vertices of the triangulation \({\mathcal T}\) by
where \({\mathcal N}_I\) is the set of all interior vertices, and \({\mathcal N}_D\) and \({\mathcal N}_N\) are the sets of vertices on \({\bar{\Gamma }}_D\) and \(\Gamma _N\), respectively. Denote the set of all edges of the triangulation \({\mathcal T}\) by
where \({\mathcal E}_I\) is the set of all interior edges and \({\mathcal E}_D\) and \({\mathcal E}_N\) are the sets of boundary edges on \(\Gamma _D\) and \(\Gamma _N\), respectively. For each element \(K \in {\mathcal T}\), denote by \(h_K\) the diameter of K, and by \({\mathcal N}_K\) and \({\mathcal E}_K\) the sets of all vertices and edges on K, respectively.
For simplicity, we assume that the interface cuts the partition with the following properties:
- (I):
If \(\Gamma \) meets an edge at more than one point, then this edge is part of \(\Gamma \).
- (II):
If the case (I) does not occur, then \(\Gamma \) must intersect a triangle at two points, and these two points must be on different edges of this triangle.
Based on the above assumptions, all triangular elements in the partition can be categorized into two classes: non-interface elements that either has no intersection with \(\Gamma \) or \(\Gamma \cap K \subset \partial K\), and interface elements whose interior is cut through by \(\Gamma \).
Denote the set of all interface elements by \({\mathcal T}^{int}\). For each interface triangle K we let \(\Gamma _K = \Gamma \cap K\) and \({\tilde{\Gamma }}_K\) be the line segment approximating \(\Gamma _K\) by connecting two endpoints of \(\Gamma _K\). Let \(K^+ = K \cap \Omega ^+\) and \(K^- = K \cap \Omega ^-\). Also we let \({\tilde{K}}^+\) and \({\tilde{K}}^-\) be the two sub-elements of K separated by \({\tilde{\Gamma }}_K\). From the setting above, it is easy to see that \({\tilde{K}}^\pm \) is either a triangle or a quadrangle. Also we define
which is the region enclosed by \(\Gamma _K\) and \({\tilde{\Gamma }}_K\). Under the assumption that \(\Gamma _K\) is \(C^2\)-continuous, the area of the \(S_K\) is of \(O(h^3_K)\) [21]. Figure 2 provides an illustration of a typical triangular interface element.
A triangular interface element
For an edge \(F \in {\mathcal E}\), if F is cut through by \(\Gamma \), i.e., \(F \cap \Gamma \ne \emptyset \) and \(F \not \subset \Gamma \), then F is called an interface edge and denote by \({{{\mathcal E}}^{int}}\) the set of all such interface edges. For each \(F \in {\mathcal E}\), denote by \(h_F\) the length of F. Denote by \(\mathbf{n}_F =(n_1, n_2)\) and \(\mathbf{t}_F=(-n_2, n_1)\) the unit vectors normal and tangential to F, respectively. Let \(K_{F,1}\) and \(K_{F,2}\) be the two elements sharing the common edge \(F \in {\mathcal E}_I\) such that the unit outward vector normal to \(K_{F,1}\) coincides with \(\mathbf{n}_F\). When \(F\subset \partial \Omega \), \(\mathbf{n}_F\) is the unit outward vector normal to \(\partial \Omega \), and denote by \(K_{F,1}\) the boundary element having the edge F. For a function v that is defined on \(K_{F,1} \cup K_{F,2}\), denote its traces on F by \(v|_F^1\) and \(v|_F^2\) restricted on \(K_{F,1}\) and \(K_{F,2}\), respectively. Define the jump of a function v on the edge F by
and the average of a function v on the edge F by
It is easy to verify that
For simplicity, we may drop the subscript F in the notations \([\![\cdot ]\!] _F\) and \(\{\cdot \}_F\) if there is no confusion on where the jump and average are defined.
2.2 IFEM Approximation
For simplicity, we assume that the interface does not intersect with the boundary, i.e., \({{{\mathcal E}}^{int}}\subset {\mathcal E}_I\). Let \({\tilde{\alpha }}\) be an approximation of \(\alpha \) such that
For each interface element \(K \in {\mathcal T}^{int}\), define the local IFE space by
where \(P_1(w)\) is the space of all polynomial functions in w of degree no more than 1. The global IFE space \(\mathcal {S}({\mathcal T})\) is then defined to include all functions such that
- 1.
\(v|_K \in {\tilde{P}}_1(K)\) for all \(K \in {\mathcal T}^{int}\) and \(v|_K \in P_1(K)\) for all \(K \in {\mathcal T}/{\mathcal T}^{int}\), and
- 2.
v is continuous at every vertex \(z \in {\mathcal N}\).
Note that for each \(z \in {\mathcal N}\), there exists a unique IFE nodal basis function [21, 24], denoted by \({\tilde{\lambda }}_z \in \mathcal {S}({\mathcal T})\), such that
where \(\delta \) is the Kronecker delta function.
The partially penalized IFEM solution for the interface problem is to find \(u_{\mathcal T}\in \mathcal {S}_D({\mathcal T}) = \left\{ v \in \mathcal {S}({\mathcal T}) \,:\, v =0 \text{ on } \Gamma _D \right\} \) such that
where the bilinear form \(a_h(w,v )\) is defined by
Here \(\epsilon \) may take the values \(-1\), 0, and 1, corresponding to symmetric, incomplete, and non-symmetric IFEM. The constant \(\gamma \) is the stability parameter and needs to be chosen large enough for symmetric and incomplete IFEMs to guarantee the coercivity. For non-symmetric IFEM, the constant \(\gamma \) is only required to be positive. For more details on the partially penalized IFEM, we refer readers to [25].
Remark 2.1
By the definition of \({\tilde{\alpha }}\) and \({\tilde{\Gamma }}_K\) it is easy to see that \({\tilde{\alpha }} = \alpha \) on all \(F \in {\mathcal E}^{int}\).
2.3 Inconsistency Error
Due to the geometrical approximation of the interface curve \(\Gamma \) by a polygonal interface \({\tilde{\Gamma }} = \bigcup \limits _{K\in {\mathcal T}^{int}} {\tilde{\Gamma }}_K\), the following geometrical inconsistent error exists. By (2.6) and integration by parts we have for any \(v \in \mathcal {S}_D({\mathcal T})\)
Remark 2.2
If the interface \(\Gamma \) is a polygon such that \({\tilde{\alpha }} = \alpha \) on each interface element, then the term (2.7) vanishes. In this case, the partially penalized IFEM scheme is consistent. In general, for a curved interface \(\Gamma \), the global convergence of IFEM will not be affected by such linear approximation of the interface, since the partially penalized IFEM also uses piecewise linear approximation [7].
3 Residual-Based A Posteriori Error Estimation
In this section, we introduce the residual-based error estimator for the partially penalized IFEM. We note that the classical residual-based a posteriori error estimation for conforming finite element methods on fitted meshes consists of element residual and the jump of the normal flux on edges. For the IFEM, it is also necessary to include the jump of the tangential derivative on interface edges since the IFEM solution may not be continuous across interface edges.
Define the normal flux jump of \(u_{\mathcal T}\) on each edge by
and the tangential derivative jump of \(u_{\mathcal T}\) on each edge by
Note that on each interface edge \(F \in {{{\mathcal E}}^{int}}\) both \(j_{n,F}\) and \(j_{t,F}\) are piecewise constant.
For all \(K \in {\mathcal T}\) we define the local error indicator \(\eta _K\) by
where \({\tilde{\alpha }}_F(x)=\max \left( {\tilde{\alpha }}|_F^1(x), {\tilde{\alpha }}|_F^2(x)\right) \), and \(S_K\) is defined in (2.4). Note that \({\tilde{\alpha }}_F(x)\) is a constant on F when \(F \notin {{{\mathcal E}}^{int}}\). The global error estimator \(\eta \) is then defined by
Remark 3.1
If K is a non-interface element, i.e., \({\mathcal E}_K \cap {{{\mathcal E}}^{int}}= \emptyset \), the first and second terms in (3.1) vanish. In this case, the local error indicator \(\eta _K\) is identical to the residual-based error indicator for the classical body-fitting conforming finite element method [6].
4 Global Reliability
In this section, we establish the reliability bound of the global estimator \(\eta \) given in (3.2). For each \(z \in {\mathcal N}\), let \(\omega _z\) be the union of all elements sharing z as a common vertex. To this end, let \({\mathcal N}_\Gamma \) be the set of all vertices z such that \(\text{ meas }_{d-1} (\omega _z \cap \Gamma ) >0 \), and define
where \(\alpha _z= \alpha (x)|_{\omega _z}\) and \(f_z\) is the average value of f on \(\omega _z\).
Remark 4.1
The first term in \(H_f({\mathcal T})\) is a higher order term for \(f \in L^2(\Omega )\) [12]. It is also well known that for linear finite element methods the edge residuals are dominant. In our adaptive algorithm the element residual is also omitted.
4.1 Helmholtz Decomposition
Let
For \(\phi \in H^1(\Omega )\), define the adjoint curl operator by \(\nabla ^{\perp }\, \phi = \left( - \dfrac{\partial \phi }{\partial y}, \,\dfrac{\partial \phi }{\partial x} \right) \). For each \(v \in \mathcal {S}({\mathcal T})\), we define the discrete gradient operator \(\nabla _h\) by
Lemma 4.1
(Helmholtz Decomposition) Let u and \(u_{\mathcal T}\) be the solutions of (2.3) and (2.6), respectively. Then there exist uniquely \(\phi \in H_{D}^1(\Omega )\) and \(\psi \in H_N^1(\Omega )\) such that
Moreover,
Proof
The proof can be referred to [3, 15]. Here we also sketch a proof for the convenience of readers. Let \({\tilde{\mathbf{e}}}_\sigma := \alpha \nabla u - {\tilde{\alpha }} \nabla _h u_{\mathcal T}\) and \(e_{\mathcal T}= u - u_{\mathcal T}\). Assume that \(\phi \in H^1(\Omega )\) solves the following equation:
with the boundary conditions
Then there exists \(\psi \in H^1(\Omega )\) such that
Moreover, we have \( \dfrac{\partial \psi }{\partial t}= \nabla ^\perp \psi \cdot \mathbf{n}=({\tilde{\mathbf{e}}}_\sigma - \alpha \nabla \phi ) \cdot \mathbf{n}=0\) on \(\Gamma _N\), hence, \(\psi \in H_N^1(\Omega )\). By integration by parts and the boundary conditions it is easy to check that
This completes the proof of the lemma. \(\square \)
Lemma 4.2
Let \(\phi \) and \(\psi \) be given in (4.1). Then we have the following error representations in the weighted semi-\(H^1\) norm:
for any \(v \in \mathcal {S}_D({\mathcal T})\) and
Proof
Let \(v \in \mathcal {S}_D({\mathcal T})\) be arbitrary. Applying (4.2), (4.1), and integration by parts gives
Applying integration by parts again gives
The last equality used (2.5), \(\phi -v =0\) on \(\Gamma _D\), and the facts that
By integration by parts and (2.6) we also have
which, together with (4.5) and (4.6), gives (4.3).
To prove (4.4), by (4.2), (4.1), integration by parts, and the facts that
we have
Hence, we obtain (4.4). This completes the proof of the lemma. \(\square \)
4.2 Modified Clément-Type Interpolation
Define a modified Clément-type interpolation operator \(\mathcal {I}_h\,:\, H_{D}^1(\Omega ) \rightarrow S_{D}({\mathcal T})\) by
where \(\pi _z\) is defined by
where \(\lambda _z \) and \({\tilde{\lambda }}_z\) are the classical barycentric hat function and the linear IFEM nodal basis function of \(\mathcal {S}({\mathcal T})\) associated to z, respectively. Note that
By Lemma 6.1 in [12], there holds
The following lemma provides the approximation and stability properties of the modified Clément-type interpolation operator.
Lemma 4.3
(Clément-type Interpolation) Let \(v \in H_{D}^1(\Omega )\), and \(\mathcal {I}_h v \in \mathcal {S}_D({\mathcal T})\) be the interpolation of v defined in (4.8). Then there exists a constant \(C>0\) that is independent of the mesh size and the location of the interface such that
where \(\omega _K\) is the union of all elements sharing at least one vertex with K.
Proof
Without loss of generality, assume that \(K \in {\mathcal T}\) is an interior element. To prove (4.11), by the partition of unity, the triangle inequality, the boundedness of IFE basis functions \(\Vert {\tilde{\lambda }}_z\Vert _{\infty ,K} \le C(\rho )\) (Theorem 2.4, [21]), and (4.10), we have
To prove (4.12), by the partition of unity, the triangle inequality, the fact that \(\Vert \nabla {\tilde{\lambda }}_z\Vert _{\infty ,K} \le C h_K^{-1}\) (Theorem 2.4, [21]), and (4.10), we have
Finally, (4.13) follows from the partition of unity, the triangle inequality, the trace inequality, and (4.10):
This completes the proof of the lemma. \(\square \)
Lemma 4.4
There exists a constant \(C>0\), independent of the mesh size and the location of the interface, such that
and
Proof
We first prove the results on the reference element K formed by vertices \(((0,0), (1,0), (0,1))\) and let F be the edge of K on the x-axis. Without loss of generality, let \((a,0), 0< a < 1\), be the interface point on F and \([\![u_{\mathcal T}]\!] _F(a,0) =b\). Note that \([\![u_{\mathcal T}]\!] _F\) takes the value 0 at both endpoints (0, 0) and (1, 0). By direct calculations, we have
Regarding the \(H^{1/2}\)-norm, we have
where \({\tilde{H}}^1(K) = \{v \in H^1(K) : v = [\![u_{\mathcal T}]\!] \,~\text{ on }\, F, ~v = 0\, ~\text{ on } \,\partial K {\setminus } \{F \}\). In particular, let
where \(K_1\) is the subtriangle fromed by ((0, 0), (a, 0), (0, 1)) and \(K_2 = K {\setminus } K_1\). It is easy to verify that \(v \in {\tilde{H}}^1(K)\). Another direct calculation gives
It is easy to verify that
which, together with the scaling argument, gives (4.14) and (4.15). This completes the proof of the lemma. \(\square \)
Lemma 4.5
Let \(\phi \) be given in (4.1). There exists a constant C independent of the mesh size and the location of the interface such that
Proof
By Lemma 4.2,
Let \(v= \phi _I \in \mathcal {S}_D({\mathcal T})\) be the modified Clément-type interpolation defined in (4.8) of \(\phi \). By the partition of unity, the fact that \({\tilde{\lambda }}_z = \lambda _z\) for \(z \in {\mathcal N}{\setminus } {\mathcal N}_\Gamma \), the Cauchy–Schwarz inequality, (4.9), and (4.10), we have
For any \(F \in ({\mathcal E}_I \cup {\mathcal E}_N) {\setminus } {{{\mathcal E}}^{int}}\), it follows from the continuity of \(\phi -\phi _{I}\) on F, Cauchy–Schwarz inequality, and (4.13) that
For any \(F \in {{{\mathcal E}}^{int}}\), it then follows from the Cauchy–Schwarz and Young’s inequality and (4.13) that
where \(\omega _F = \omega _{K_{F,1}} \cup \omega _{K_{F,2}}\), which, together with (4.18) and the Cauchy–Schwarz inequality, yields
To bound \(I_3\), we apply the following trace inequality for functions in the IFEM space (see Lemma 3.2 in [25]). Let \(v \in \mathcal {S}_D({\mathcal T})\) be arbitrary, then
It now follows from the Cauchy–Schwarz inequality, (4.20), and (4.12) that
Together with (4.14) we obtain
To bound \(I_4\), we use the fact that
the Cauchy–Schwarz and triangle inequalities, and (4.13) to obtain
which, combining with (4.14), yields
Finally, (4.16) is a direct consequence of (4.17), (4.19), (4.21), (4.22) and the Young’s inequality. This completes the proof of the lemma. \(\square \)
Lemma 4.6
Let \(\psi \) be given in (4.1). There exists a constant C independent of the mesh size and the location of the interface such that
Proof
For the first term in (4.4) it follows from the duality inequality, (4.15), the trace and Young’s inequalities that
For the second term in (4.4) it follows from the Cauchy–Schwarz inequality that
Combining (4.24)–(4.25) and (4.4) gives (4.23). This completes the proof of the lemma. \(\square \)
Theorem 4.7
(Global Reliability) There exists a constant \(C_r>0\) that is independent of the location of the interface and the mesh size, such that
Proof
First by adding and subtracting proper terms we have
Applying the Cauchy–Schwarz and Young’s inequalities also gives
Equation (4.26) is then a direct result of (4.27)–(4.29), Lemmas 4.5, 4.6 and the Young’s inequality. This completes the proof of theorem. \(\square \)
5 Local Efficiency
In this section, we establish the efficiency bound for the error indicators \(\eta _K\) defined in (3.1) for every element \(K \in {\mathcal T}\). For non-interface elements, the efficiency bound for \(\eta _K\) is well known (see [6]) and the key technique is using the local edge and element bubble functions. However, the same technique without modification becomes invalid for interface elements because jumps of the normal flux and the tangential derivative on the interface edges become piecewise constant. To overcome this difficulty it is natural to design more localized bubble functions that allow us to derive the efficiency bounds on \(F^+\) and \(F^-\) separately for each \(F \in {{{\mathcal E}}^{int}}\).
For each \(F \in {{{\mathcal E}}^{int}}\) we first design auxiliary element and edge bubble functions associated to \(F^-\). Choose \({\tilde{K}}_{F,1} \subset K_{F,1}\) and \({\tilde{K}}_{F,2} \subset K_{F,2}\) to be two regular triangular sub-elements such that \(F^-\) is a common edge of both \({\tilde{K}}_{F,1}\) and \({\tilde{K}}_{F, 2}\), to be more precise,
Note that \({\tilde{K}}_{F,1}\) and \({\tilde{K}}_{F,2}\) are not necessarily in \(K_{F,1}^-\) and \(K_{F,2}^-\), respectively. The key requirement here is to make sub-elements \({\tilde{K}}_{F,1}, {\tilde{K}}_{F,2}\) regular while \(K_{F,1}^-\) and \(K_{F,2}^-\) may not be in general. For each sub-element \({\tilde{K}} \in \{{\tilde{K}}_{F,1}, {\tilde{K}}_{F,2}\}\), define the auxiliary element bubble function \(\upsilon _{{\tilde{K}}}\) such that (i) \(\upsilon _{{\tilde{K}}} \in P_3({\tilde{K}})\), (ii) \(\upsilon _{{\tilde{K}}}|_{\partial {\tilde{K}}} \equiv 0\) and (iii) \(\upsilon _{{\tilde{K}}}\) takes the value 1 at the barycentric center of \({\tilde{K}}\). We also define the auxiliary edge bubble function \(\upsilon _{F^-}\) for \(F^-\) such that (i) \(\upsilon _{F^-} |_{{\tilde{K}}} \in P_2({\tilde{K}})\) for \({\tilde{K}} \in \{ {\tilde{K}}_{F,1}, {\tilde{K}}_{F,2}\}\), (ii) \(\upsilon _{F^-}|_{\partial ({\tilde{K}}_{F,1} \cup {\tilde{K}}_{F,2})} \equiv 0\) and (iii) \( \upsilon _{F^-} \) takes value 1 at the middle point of \(F^-\). It is easy to verify that \(\upsilon _{{\tilde{K}}}\) for \({\tilde{K}} \in \{ {\tilde{K}}_{F,1}, {\tilde{K}}_{F,2}\} \) and \(\upsilon _{F^-}\) uniquely exist. Let \(w_{F^-} = (j_{n,F}|_{F^-})v_{F^-} \) and \(w_{{\tilde{K}}} = v_{{\tilde{K}}}f_K\) with \(f_K\) being the average of f on K.
Applying the continuity of \(\alpha \nabla u \cdot \mathbf{n}_F\), the divergence theorem, and the Cauchy–Schwarz inequality gives
which, combining with the properties of \(w_{F^-}\) that
yields
We now establish the efficiency bound for the element residual \(\Vert f\Vert _{0,{\tilde{K}}}\) for \({\tilde{K}} \in \{ {\tilde{K}}_{F,1}, {\tilde{K}}_{F,2}\}\). By the property of \(w_{{\tilde{K}}}\), the triangle inequality, the divergence theorem, and the Cauchy–Schwarz inequality, we have
which, combining with similar properties for \(w_{{\tilde{K}}}\):
gives
Adding proper weights gives
where
and the constant C is independent of the mesh size and the location of the interface but may depend on \(\rho \). Similarly, one can prove that
and, after adding proper weights, that
Similarly, by defining auxiliary bubble functions on \(F^+\), we also have the following local efficiency results on \(F^+\):
and
For the first case we assume that for each \(F \in {{{\mathcal E}}^{int}}\), there exist positive constants \(\lambda \) and \(\Lambda \) such that
Lemma 5.1
Let u and \(u_{\mathcal T}\) be the solution in (2.3) and (2.6), respectively. Then there exists a constant \(C_1\) that is independent of the mesh size but may depend on \(\lambda , \Lambda ,\) and \( \rho \) such that
Proof
In the case that K is not an interface element, (5.8) is a well known result (see, e.g., [8]). In the case that K is an interface element, by (5.3) and (5.5), we have
where \(C_1\) is independent of the mesh size but might depend on \(\rho \), \(\lambda \), and \(\Lambda \). Similarly, by (5.4) and (5.6), we have
where \(C_1\) is independent of the mesh size but might depend on \(\rho \), \(\lambda \), and \(\Lambda \). This completes the proof of the lemma. \(\square \)
In the above lemma, the efficient constant \(C_1\) blows up when the \(\lambda \) and \(\Lambda \) become extreme. However, we note that in the extreme circumstances the related basis functions for IFEM are very “close” to classical FE nodal basis functions thanks to its consistency with standard FE basis functions (see [21] for more detail). The partially penalized IFEM solution then should be also “close” to the classical finite element solution on fitted meshes. In the following lemma, we prove the efficiency in a different approach that yields a bounded coefficient constant when the \(\lambda \) and \(\Lambda \) are extreme.
Lemma 5.2
There exists a constant \(C_2\) that is independent of the mesh size and the location of the interface such that for each interface edge \(F \in {\mathcal E}^{int}\) the following efficiency bound holds:
Proof
Without loss of generality assume that \(h_F^+ \ge h_F^-\). Let \(K \in \{K_{F,1}, K_{F,2} \}\). Define \({\hat{K}} \subset K\) to be the triangle such that (i) \(F^+\) is a complete edge of \({\hat{K}}\) and (ii) the vertex in K that is opposite to F, denote by \(z_{F,K}\), is also a vertex of \({\hat{K}}\). It is obvious that \({\hat{K}}\) is regular. Also define \({\hat{u}}_K\) to be a piecewise linear function such that \({\hat{u}}_K \equiv u_{\mathcal T}\) on \(K^+\) (or \({\hat{K}}\)), if \({\hat{K}} \subset K^+\) (or \(K^+ \subset {\hat{K}}\)) and that \({\hat{u}}_K|_F\) is linear. It is obvious that \({\hat{u}}_K\) uniquely exists. Finally define \({\hat{u}}_{{\mathcal T}}\) such that \({\hat{u}}_{{\mathcal T}}|_K = {\hat{u}}_K\), \(K \in \{K_{F,1}, K_{F,2}\}\). By the triangle inequality, we have
Then by the definition of \({\hat{u}}_{\mathcal T}\) and a direct computation, we have for each \(K \in \{K_{F,1}, K_{F,2}\}\):
where \(\rho _{K^-}\) denotes that radius of the largest ball inscribed in \(K^-\). This, combining with the fact that \(|(u_{\mathcal T}- {\hat{u}}_{\mathcal T})(z_F^-)| \lesssim \rho _{K^-}\) (See [21, 24] for the consistence of FE and IFE functions when one piece of the interface element is small.) yields
Now applying the fact that \([\![D_n {\hat{u}}_{\mathcal T}]\!] |_F\) is a constant and the standard efficiency result on \({\hat{K}}_{F,1}\) and \({\hat{K}}_{F,2}\) gives
where the constant \(C_2\) is independent of the mesh size and the location of the interface.
Similarly we can prove that
where the constant \(C_2\) is also independent of the mesh size and the location of the interface. Finally (5.10) is a direct result of (5.15) and (5.16). \(\square \)
Remark 5.1
(5.15) indicates that for the case when \(\lambda \) and \(\Lambda \) are extreme the term \(h_F^{1/2}\sqrt{h_{F}^-}\) becomes negligible, and, therefore, \(C_2\) can be used as an effective efficiency constant.
Theorem 5.3
Let u and \(u_{\mathcal T}\) be the solution in (2.3) and (2.6), respectively. The following efficiency bound holds for any \(K \in {\mathcal T}^{int}\):
where
\(\chi \) is the characteristic function and \(C_1\) and \(C_2\) are the efficiency constants in Lemma 5.1 and Lemma 5.2, respectively.
6 Numerical Results
In this section, we report some numerical results to demonstrate the performance of the residual-based error estimator for partially penalized IFEM.
For the first three examples, we consider a diffusion interface problem with a smooth elliptical interface curve which has been reported in [25, 27]. Let \(\Omega =[-1,1]^2\), and the interface \(\Gamma \) is an ellipse centered at \((x_0,y_0) = (0,0)\) with horizontal semi-axis \(a= \frac{\pi }{6.28}\) and the vertical semi-axis \(b = \frac{3}{2}a\). The interface separates \(\Omega \) into two sub-domains, denoted by \(\Omega ^-\) and \(\Omega ^+\) such that
where
The exact solution to this interface problem is
Here \(\beta ^\pm >0\) are the diffusion coefficients, and \(p>0\) is the regularity parameter. In the following, we use \(\rho =\frac{\beta ^+}{\beta ^-}\) to denote the ratio of the coefficient jump.
Our adaptive mesh refinement follows the standard procedure:
We solve the interface problem using IFEM in (2.6), then we compute the residual-based error indicator \(\eta _K\) on each element by (3.1). We adopt the equilibration marking strategy, i.e., construct a minimal subset \(\hat{\mathcal {T}}\) of \(\mathcal {T}\) such that
where the threshold \(\theta =0.5\). Finally we refine the marked triangles by newest vertex bisection [28]. The initial mesh is formed by first partitioning the domain into a \(4\times 4\) congruent rectangles, then cutting each rectangle into two congruent triangles by connecting its diagonal with positive slope.
Example 6.1
(Piecewise smooth solution with a moderate jump) In this example, we let \(\rho = 100\) and \(p=5\) which represents a moderate coefficient contrast and piecewise smooth solution. In Fig. 3, we list, from left to right, some typical meshes of a similar number of elements and degrees of freedom (DOF) generated by the uniform IFEM, the adaptive IFEM, and the adaptive FEM on unfitted mesh [13]. We observe that there is not much local mesh refinement around the interface for the adaptive IFEM in the middle of Fig. 3, compared with the uniform mesh in the left plot. This indicates that the errors of the partially penalized IFEM on uniform mesh are almost equally distributed, and IFEM itself can resolve the interface accurately for moderate coefficient jump. Whereas using the finite element method on unfitted meshes requires much more local mesh refinement around the interface (see the plot on the right of Fig. 3) to resolve the non-smooth feature of interface problems.
Convergence of uniform IFEM, adaptive IFEM, and adaptive FEM for Example 6.1
In Fig. 4, we report the convergence of these three methods and the residual-based error estimator for the adaptive IFEM. The slopes of the log(DOF)-log\((\Vert \alpha ^{1/2}(\nabla u-\nabla _h u_{\mathcal T})\Vert _{0,\Omega } )\) and the log(DOF)-log(\(\eta \)) for the adaptive IFEM are both very close to \(-1/2\), which indicates the optimal-order decay of errors with respect to the number of unknowns and, hence, the efficiency of our local error indicators. We use the following efficiency index,
to test the efficiency of our residual-based error estimator. The eff-index is very stable at every mesh level and the value is around 3. We note that the errors of uniform IFEM are very close to the errors of adaptive IFEM. This again indicates that the IFEM itself can resolve the interface accurately for moderate coefficient jumps and piecewise smooth solutions. However, using the standard FEM, the magnitudes of errors are much larger than those of IFEM, with similar degrees of freedom.
Example 6.2
(Piecewise smooth solution with a large jump) In this example, we test the large jump case by choosing \(\rho =10^6\). In this case, the true solution possesses a very steep gradient at the interface.
The left plot of Fig. 5 shows a typical mesh for the adaptive IFEM, which, compared with Fig. 3, has much denser refinement around the interface. In the right plot of Fig. 5, we observe the optimal-rate decay of the errors and the estimators. Nevertheless, even if the convergence rate is optimal for the uniform IFEM, the magnitudes of errors are significantly larger than the errors of adaptive IFEM. Hence, applying adaptive mesh refinement is computationally more efficient for interface problems with large coefficient jump even for IFEM. The efficiency indices are between 2.5 and 3.5 on all meshes except the first few coarse ones, and they become more stable (close to 3) as the computations reach the asymptotical convergence region. This phenomenon indicates the robustness of the error estimation with respect to the ratio of the coefficient jump.
Furthermore, the numerical solutions and the error surfaces on the uniform and adaptive meshes with similar DOFs are depicted in Figs. 6 and 7, respectively. We can observe that the error is significantly diminished for the adaptive solution.
Mesh generated by the adaptive IFEM (left) and the convergence of adaptive and uniform IFEM (right) for Example 6.2
Numerical solutions of uniform (left) and adaptive (right) IFEM with similar DOFs for Example 6.2
Point-wise errors of uniform (left) and adaptive (right) IFEM with similar DOFs for Example 6.2
Example 6.3
(Solution with singularity and a large coefficient jump) In this example, we consider the interface problem with a large coefficient jump and a solution singularity. We choose \(\rho =10^6\) and \(p=0.5\) in the exact solution (6.1). Note that the solution is merely in \(H^{1.5-\epsilon }(\Omega )\) for any \(\epsilon >0\) and the solution becomes singular at the origin.
The left plot of Fig. 8 shows a typical mesh of the adaptive IFEM, and we observe that the mesh is densely refined around the interface as well as the point of singularity. The right plot of Fig. 8 shows the optimal-rate decay of the errors and the estimators of our adaptive IFEM. Again, the averaging effectivity index is close to 3, which is similar to that in previous examples. This indicates the uniform effectivity of the error estimate with respect to the type of elements, i.e., interface and non-interface elements. The comparison of error in the right plot of Fig. 8 shows a stronger superiority of the adaptive mesh refinement than the uniform mesh refinement. In fact, the convergence of IFEM with uniform mesh refinement is not optimal, due to the singularity of the solution.
The numerical solutions and error surfaces for the adaptive IFEM and uniform IFEM are depicted in Figs. 9 and 10, respectively. It is easy to see that the numerical solution on the uniform mesh cannot resolve the behavior of exact solution accurately at the singularity point, and the error of the uniform solution possesses a very high peak around the singular point, while we can barely observe this phenomenon from the adaptive solution.
Mesh generated by the adaptive IFEM (left) and the convergence of adaptive and uniform IFEM (right) for Example 6.3
Numerical solutions of uniform (left) and adaptive (right) IFEM with similar DOFs for Example 6.3
Point-wise errors of uniform (left) and adaptive (right) IFEM with similar DOFs for Example 6.3
Example 6.4
(Solution with complicated interface shape) In this example, we consider an interface problem with a more complicated interfacial shape. The exact solution has a petal-shaped interface and it is defined through the following level set function:
where
Due to the complexity of interface shape, we start the AMR procedure with a finer initial mesh, a \(16\times 16\) Cartesian triangular mesh. A typical mesh is depicted on the left plot of Fig. 11. Comparing with Example 6.1, in which the interface is an ellipse (see Fig. 3), the refinement around the interface is denser. This is because the larger curvature of the interface causes the larger value of the inconsistency term in the error indicator in (3.1).
The convergence plot depicted in the right plot of Fig. 11 indicates the optimal-rate decay for both the errors and estimators. The errors of the adaptive solution are a little smaller than the errors of the uniform solution with similar degrees of freedom, although the latter also converge in an optimal rate. Moreover, the efficient index for this example is close to 3 which is similar to those in the previous examples. Numerical solutions and error surfaces of uniform and adaptive IFEMs are reported in Figs. 12 and 13, respectively. We can again see that the errors of adaptive IFEM are smaller than the errors of uniform IFEM has given similar degrees of freedom.
Mesh generated by the adaptive IFEM (left) and the convergence of adaptive and uniform IFEM for Example 6.4
Numerical solutions of uniform (left) and adaptive (right) IFEM with similar DOFs for Example 6.4
Point-wise errors of uniform (left) and adaptive (right) IFEM for Example section 6 with similar DOFs for Example 6.4
Example 6.5
(Effect of the inconsistent error terms) In this example, we will explore the effect of the inconsistent error term for different interface geometries by revisiting the Examples 6.1 and 6.4 which have a simple ellipse interface and a more complicated petal interface. We consider the following error indicator
which is same as \(\eta _K^2\) in (3.1) but without the inconsistent error term \(\Vert {\tilde{\alpha }}^{1/2} \nabla u_{\mathcal T}\Vert _{S_K}^2\). The global error estimator is defined in the standard way:
First, we compare the convergences using the error estimator \(\xi \) and \(\eta \) for Example 6.1. In the left plot of Fig. 14, the estimators \(\xi \) and \(\eta \) are very similar at all meshes, and the errors of the corresponding IFEM solutions guided by these two estimators are also close. We believe that due to the simple and smooth interface shape of this example (an ellipse interface), the inconsistent error term is negligible. Next, we use the new error estimator \(\xi \) in Example 6.4 In which the geometry of interface is more complicated. As we can see in the right plot of Fig. 14, the error estimators \(\eta \) and \(\xi \) and the corresponding IFEM solutions show notable differences, especially on the first few coarse meshes. In this sense, including the geometrical correction (inconsistent error) term leads to better error indication particularly on coarse meshes. We also note that as the mesh is adaptively refined, the error estimators \(\eta \) and \(\xi \) become closer, as well as the IFEM solutions leading by these two estimators.
Comparison of estimators \(\eta \) and \(\xi \) for Example 1 (left) and Example 4 (right)
Example 6.6
(Additional Comments on Large Jump Scenarios) In this test, we revisit the large jump scenario using the test problem from Example 6.2. To show that the necessity of performing adaptive mesh strategy for IFEM, we use the true error in energy norm as our error indicator, i.e.,
and the global error estimator is the true error in the energy norm, i.e.,
An adaptive mesh using the error estimator \(\eta ^*\) with a similar number of triangles are shown in the left plot of Fig. 15. In both cases, the refinement is concentrated around the interface, which is similar to the adaptive mesh in Fig. 5. This again shows that the partially penalized IFEM itself may not be sufficient to obtain an accurate solution for extremely large jumps of the coefficient. In this case, the adaptive mesh refinement is more advantageous. The right plot of Fig. 15 shows the convergence of the errors governed by the estimator \(\eta \) and the true error \(\eta ^*\). We can see that both converge at an optimal rate, although using the true error as the estimator gives slightly more accurate solutions.
Mesh generated by the adaptive IFEM (left) using true error as the error indicators and convergence of IFEM solutions guided by \(\eta \) and by the true error \(\eta ^*\) for Example 6.2
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The research of Xu Zhang was partially supported by the National Science Foundation Grant DMS-1720425.
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He, C., Zhang, X. Residual-Based a Posteriori Error Estimation for Immersed Finite Element Methods. J Sci Comput 81, 2051–2079 (2019). https://doi.org/10.1007/s10915-019-01071-5
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DOI: https://doi.org/10.1007/s10915-019-01071-5