Convergence and Quasi-Optimality of Adaptive FEM
with Inhomogeneous Dirichlet Data
We consider the solution of a second order elliptic PDE with inhomogeneous Dirichlet data by means of adaptive lowest-order FEM. As is usually done in practice, the given Dirichlet data are discretized by nodal interpolation. As model example serves the Poisson equation with mixed Dirichlet-Neumann boundary conditions. For error estimation, we use an edge-based residual error estimator which replaces the volume residual contributions by edge oscillations. For 2D, we prove convergence of the adaptive algorithm even with quasi-optimal convergence rate. For 2D and 3D, we show convergence if the nodal interpolation operator is replaced by the -projection or the Scott-Zhang quasi-interpolation operator. As a byproduct of the proof, we show that the Scott-Zhang operator converges pointwise to a limiting operator as the mesh is locally refined. This property might be of independent interest besides the current application. Finally, numerical experiments conclude the work.
Key words and phrases:adaptive finite element methods, convergence analysis, quasi-optimality, inhomogeneous Dirichlet data
2000 Mathematics Subject Classification:65N30, 65N50.
1.1. Model problem
By now, the thorough mathematical understanding of convergence and quasi-optimality of -adaptive FEM for second-order elliptic PDEs has matured. However, the focus of the numerical analysis usually lies on model problems with homogeneous Dirichlet conditions, i.e. in with on , see e.g. [14, 15, 22, 24, 30]. On a bounded Lipschitz domain in with polygonal boundary , we consider
with mixed Dirichlet-Neumann boundary conditions. The boundary is split into two relatively open boundary parts, namely the Dirichlet boundary and the Neumann boundary , i.e. and . We assume the surface measure of the Dirichlet boundary to be positive , whereas is allowed to be empty. The given data formally satisfy , , and . As is usually required to derive (localized) a posteriori error estimators, we assume additional regularity of the given data, namely , , and .
Whereas certain work on a posteriori error estimation for (1) has been done, cf. [5, 28], none of the proposed adaptive algorithms have been proven to converge. While the inclusion of inhomogeneous Neumann conditions into the convergence analysis seems to be obvious, incorporating inhomogeneous Dirichlet conditions is technically more demanding and requires novel ideas. First, discrete finite element functions cannot satisfy general inhomogeneous Dirichlet conditions. Therefore, the adaptive algorithm has to deal with an additional discretization of . Second, this additional error has to be controlled in the natural trace space which is the fractional-order Sobolev space . Since the -norm is non-local, the a posteriori error analysis requires appropriate localization techniques. These have recently been developed in the context of adaptive boundary element methods [3, 11, 12, 16, 17, 21]: Under certain orthogonality properties of , the natural trace norm is bounded by a locally weighted -seminorm . Here, is the local mesh-width, and denotes the arclength derivative. Finally, in contrast to homogeneous Dirichlet conditions , we loose the Galerkin orthogonality in energy norm. This leads to certain technicalities to derive a contractive quasi-error which is equivalent to the overall Galerkin error in . In conclusion, quasi-optimality and even plain convergence of adaptive FEM with non-homogeneous Dirichlet data is a nontrivial task. To the best of our knowledge, only  analyzes convergence of adaptive FEM with inhomogeneous Dirichlet data. While the authors also consider the 2D model problem (1) with and lowest-order elements, their analysis relies on an artificial non-standard marking criterion. Quasi-optimal convergence rates are not analyzed and can hardly be expected in general .
It is well-known that the Poisson problem (1) admits a unique weak solution with on in the sense of traces which solves the variational formulation
Here, the test space reads , and denotes the respective -scalar products.
For the Galerkin discretization, let be a regular triangulation of into triangles . We use lowest-order conforming elements, where the ansatz space reads
Since a discrete function cannot satisfy general continuous Dirichlet conditions, we have to discretize the given data . According to the Sobolev inequality on the 1D manifold , the given Dirichlet data are continuous on . Therefore, the nodal interpoland of is well-defined. As is usually done in practice, we approximate . Again, it is well-known that there is a unique with on which solves the Galerkin formulation
Here, the test space is given by .
1.3. A posteriori error estimation
An element-based residual error estimator for this discretization reads
with corresponding refinement indicators
where denotes the jump across edges. We prove reliability and efficiency of (Proposition 2) and discrete local reliability (Proposition 3). Inspired by , we introduce an edge-based error estimator which reads
For an edge , its local contributions read
Here, denotes the edge patch, and denotes the corresponding integral mean. The advantage of is that the volume residual terms in (6) are replaced by the edge oscillations , which are generically of higher order. The choice of to measure the contribution of the Dirichlet data approximation is influenced by the Dirichlet data oscillations, cf. Section 3.1 below. We prove that and are locally equivalent (Lemma 4) and thus obtain reliability and efficiency of (Proposition 5) as well as discrete local reliability (Proposition 6).
1.4. Adaptive algorithm
We use the local contributions of to mark edges for refinement in a realization (Algorithm 7) of the standard adaptive loop (AFEM)
Our adaptive algorithm use variants of the the well-studied Dörfler marking  to mark certain edges for refinement. Throughout, we use newest vertex bisection, and at least marked edges are bisected. Given some initial mesh , the algorithm generates successively locally refined meshes with corresponding discrete solutions of (4).
1.5. Main results
The first main result (Theorem 14) states that the adaptive algorithm leads to a contraction
for some quasi-error quantity which is equivalent to the error estimator. In particular, this proves linear convergence of the adaptively generated solutions to the (unknown) weak solution of (2). The main ingredients of the proof are an equivalent error estimator for which we prove some estimator reduction
The second main result is Theorem 18 which states that the outcome of the adaptive algorithm is quasi-optimal in the sense of Stevenson : Provided the given data and the corresponding weak solution of (2) belong to the approximation class
the adaptively generated solutions also yield convergence order , i.e.
Here, denotes the set of all triangulations which can be obtained by local refinement of the initial mesh such that . Moreover, , and denote the data oscillations of the volume data , the Dirichlet data , and the Neumann data , see Section 3.1.
The ingredients for the proof are the observation that the proposed marking strategy is optimal (Proposition 15) and the Céa-type estimate
for the Galerkin solution in Lemma 17.
For 3D, nodal interpolation of the Dirichlet data is not well-defined. In the literature, it is proposed to discretize by use of the -projection  or the Scott-Zhang projection . Our third theorem (Theorem 21) states convergence of the adaptive algorithm for either choice in 2D as well as 3D. The proof relies on the analytical observation that, under adaptive mesh-refinement, the Scott-Zhang projection converges pointwise to a limiting operator (Lemma 19), which might be of independent interest. Finally, we stress that the same results (Thm. 14, 18, 21) hold if the element-based estimator from (5)–(6) instead of the edge-based estimator is used and if Algorithm 7 marks certain elements for refinement.
The remainder of this paper is organized as follows: We first collect some necessary preliminaries on, e.g., newest vertex bisection (Section 2.2) and the Scott-Zhang quasi-interpolation operator (Section 2.3). Section 3 contains the analysis of the a posteriori error estimators from (5)–(6) and from (7)–(8). Moreover, we state the adaptive Algorithm in Section 3.4. The convergence is shown in Section 4, while the quasi-optimality results are found in Section 5. Whereas the major part of the paper is concerned with the 2D model problem, Section 6 considers convergence of AFEM for 3D. Finally, some numerical experiments conclude the work.
Throughout, denotes a regular triangulation which is obtained by steps of (local) newest vertex bisection for a given initial triangulation . By , we denote the set of all interior nodes, respectively the set of all boundary nodes of . By , we denote the set of all edges of which is split into the interior edges and boundary edges . We restrict ourselves to meshes such that each has an interior node, i.e. . Note, that this is only an assumption on the initial mesh . We assume that the partition of into Dirichlet boundary and Neumann boundary is resolved, i.e. is split into and . Note that (resp. ) provides a partition of (resp. ).
For a node , the corresponding patch is defined by
For an edge , the edge patch is defined by
Moreover, for a given node ,
denotes the star of edges originating at .
2.2. Newest vertex bisection
Throughout, we assume that newest vertex bisection is used for mesh-refinement, see Figure 1. Let be a given mesh and an arbitrary set of marked edges. Then,
denotes the coarsest regular triangulation such that all marked edges have been bisected. Moreover, we write
if is a finite refinement of , i.e., there are finitely many triangulations and sets of marked edges such that and for all .
We stress that, for a fixed initial mesh , only finitely many shapes of triangles appear. In particular, only finitely many shapes of patches (16)–(17) appear. This observation will be used below. Moreover, newest vertex bisection guarantees that any sequence of generated meshes with is uniformly shape regular in the sense of
Further details are found in [32, Chapter 4].
2.3. Scott-Zhang quasi-interpolation and discrete lifting operator
Our analysis below makes heavy use of the Scott-Zhang projection from : For all nodes , one chooses an edge with . For , this choice is restricted to . Moreover, for , we even enforce . For , is then defined by
for a node . Here, denotes the dual basis function defined by , and denotes the hat function associated with . By definition, we then have the following projection properties
for all ,
for all and with ,
for all and with ,
i.e. the projection preserves discrete (Dirichlet) boundary data. Moreover, satisfies the following stability property
and approximation property
where depends only on . Together with the projection property onto , it is an easy consequence of the stability (22) of that
for all . In particular, is quasi-optimal in the sense of the Céa lemma with respect to and , i.e.
Moreover, allows to define a discrete lifting operator
whose operator norm is uniformly bounded in terms of . Here, denotes an arbitrary lifting operator, i.e. for all , see e.g. .
Finally, we put emphasis on the fact that our definition of also provides an operator which is consistent in the sense that for all . Using the definition of as the trace space of and the stability (22), we see
for all , i.e. is a continuous projection with respect to the -norm. In particular, also provides a continuous projection , since
for all . As before, this definition is consistent with the previous notation of since for all .
3. A Posteriori Error Estimation and Adaptive Mesh-Refinement
3.1. Data oscillations
We start with the element data oscillations
and where denotes the integral mean over an element . These arise in the efficiency estimate for residual error estimators.
Our residual error estimator will involve the edge data oscillations
Here, is the edge patch from (17), and is the corresponding integral mean of .
For the analysis, we shall additionally need the node data oscillations
Here, is the node patch from (16), and is the corresponding integral mean of .
Moreover, the efficiency needs the Neumann data oscillations
and where denotes the integral mean over an edge .
Finally, the approximation of the Dirichlet data is controlled by the Dirichlet data oscillations
Recall that, on the 1D manifold , the derivative of the nodal interpoland is the elementwise best approximation of the derivative by piecewise constants, i.e.,
According to the elementwise Pythagoras theorem, this implies
and all Dirichlet edges . This observation will be crucial in the analysis below. Moreover, (32) yields
The following result is found in [17, Lemma 2.2].
Let and let denote the nodal interpoland of on . Then,
where the constant depends only on the shape regularity constant and .∎
3.2. Element-based residual error estimator
Proposition 2 (reliability and efficiency of ).
The error estimator is reliable
The constants depend only on the shape regularity constant and on .
Sketch of proof.
We consider a continuous auxiliary problem
with unique solution . We then have norm equivalence as well as . From this, we obtain
Whereas the second term is controlled by Lemma 1, the first can be handled as for homogeneous Dirichlet data, i.e. use of the Galerkin orthogonality combined with approximation estimates for a Clément-type quasi-interpolation operator. Details are found e.g. in . This proves reliability (37).
Proposition 3 (discrete local reliability of ).
Let be an arbitrary refinement of with associated Galerkin solution . Let be the set of all elements which are refined to generate . Then, there holds
with some constant which depends only on and .
We consider a discrete auxiliary problem
with unique solution with . To estimate the -norm of in terms of the boundary data, let denote the discrete lifting operator from (26). Let be arbitrary extensions of and , respectively. Then, we have . According to the triangle inequality and a Poincaré inequality for , we first observe
Moreover, the variational formulation for yields
whence by the Cauchy-Schwarz inequality
Altogether, this proves