The convergence analysis of adaptive mixed finite element methods (AMFEM) stated in [6, 7, 11, 8]
for the Laplacian is completed in this paper for
non-selfadjoint indefinite second-order linear elliptic problems
via separate marking with the
axioms from .
Given a right-hand side
The mixed formulation of (1.1) seeks
The convergence and quasi-optimality of adaptive finite element methods for linear symmetric elliptic problems
has been discussed in the literature [1, 2, 5, 7, 8, 9, 17, 18, 20, 21] and the references mentioned therein. For the non-symmetric case
This paper develops the quasi-optimality of adaptive MFEMs
with the natural
The first intermediate solution concerns the discrete flux approximation with a prescribed divergence
on the coarse triangulation in the finer Raviart–Thomas space and a generalization of the corresponding
design from [8, 7, 11].
The second intermediate solution is the integral mean
in the proof of quasi-orthogonality (A4
The remaining parts of the paper are organized as follows. Section 2 establishes the notation, the a posteriori error estimators, the adaptive algorithm with separate marking (Safem), and recalls the axioms of adaptivity (A1)–(A4), (B1)–(B2), and quasimonotonicity (QM) with the optimal convergence rates from . Section 3 starts with the proof of stability (A1) and reduction (A2) for the error estimators and distance functions at hand. Section 4 is devoted to the discrete reliability based on a discrete Helmholtz decomposition in 2D for multiply connected domains. Section 5 verifies the quasi-orthogonality based on (1.5) with a direct proof in Lemma 1. Numerical experiments in Section 6 investigate the condition on sufficiently small parameters such as the bulk parameter and the mesh-size for optimal convergence rates.
The presentation is laid out for two-dimensional polygonal domains and the lowest-order case. The coefficients are assumed piecewise constant for simplicity to avoid extra perturbation terms as in . The generalization to 3D may follow the lines of this paper and replaces the discrete Helmholtz decomposition as in  for a simply connected domain. The analysis of higher-order finite element approximations requires a new argument for the stability of the discrete problems in that  and part of the proofs in this paper utilize the equivalence to the Crouzeix–Raviart nonconforming FEM, which is open in 3D for higher polynomial degrees.
Standard notation on Lebesgue and Sobolev spaces such as
This section first introduces the necessary notation for the definition and analysis of adaptive algorithms with separate marking. The axioms of adaptivity from  are slightly simplified to match the setting of this paper.
The piecewise gradient
2.2 A Posteriori Error Control
holds for the (squared) error estimator
Given the discrete solution
with the weighted norm from
2.3 Safem – The Adaptive Algorithm with Separate Marking
The separate marking scheme runs two alternatives A and B depending on the ratio of
The routine Refine applies the newest vertex bisection  and refines the marked triangles
2.4 Axioms and Optimal Convergence
Suppose that there exist
universal positive constants
- (A1)Stability: For all
and all ,
- (A2)Reduction: For all
and all ,
- (A3)Discrete Reliability: For all
and all ,
- (A4)Quasi-Orthogonality: For all
- (B1)Rate s Data Approximation: There exists
such that for , satisfies
- (B2)Quasimonotonicity of μ: For all
and all ,
The proof of Theorem 1 is given in  and not recalled here.
The version of this paper is even slightly simplified in that
The data approximation axioms (B1)–(B2) are discussed in , the results apply verbatim for the setting in this paper. This is exemplified in [8, Section 5] for the mixed FEM and hence not further detailed in this paper.
The analysis of (A1)–(A2) follows standard arguments and is outlined here for completeness for the problem at hand with piecewise constant coefficients with little emphasis that the global constants
There exists a universal constant
The discrete jump control plus triangle inequalities in Lebesgue spaces and in finite-dimensional Euclidean spaces
imply the stability (A1).
Throughout this section, let
Axiom (A1) holds with
The reverse triangle inequality in
Each of the terms
with the abbreviation
The mesh-size is bounded from above and so the right-hand side is bounded by the factor
Axiom (A2) holds with
For the m refined triangles
The reverse triangle inequalities in
4 Verification of (A3)
Throughout this section let
The main residual
There exists some
The initial mesh-size
A Cauchy inequality concludes the proof. ∎
The further analysis of
(Here and throughout the paper,
For the multi-connected domain Ω the decomposition of piecewise constant vector functions
is orthogonal with respect to the
The discrete Helmholtz decomposition is well known for simply-connected domains and
The modified Crouzeix–Raviart space is accompanied by a modified Raviart–Thomas space
The discrete Helmholtz decomposition allows for a characterization of the divergence-free functions.
The linear operator
is surjective and its kernel is
The divergence-free Raviart–Thomas functions in
One key argument of the reliability analysis is the split of the difference
hold for all
The solution to (4.3) is recovered from an auxiliary nonconforming problem:
for all test functions
The piecewise constant function
The proof imitates that of [4, Theorem 4.2] and generalizes it to multiply connected domains.
The arguments therein confirm the continuity of normal
components along the interior edges and prove
The first term on the right-hand side already appears in [4, p. 567, lines 1–2] and is rewritten as
This term combines with
The design of
This concludes the proof of the existence of a discrete solution to (4.3a)–(4.3c) and it remains to show the uniqueness of a discrete solution. This follows from the trivial solution to the homogeneous system
The test function
The first equation in (1.4) shows that
Lemma 5 and the modified test-function
The divergence-free term
This and the
Given any node
The combination with estimate (4.11) for
A rearrangement with the triangle-oriented error estimator concludes the proof. ∎
from Lemma 2 for some
(i) The representation of
On the other hand,
(ii) Observe from (4.3b) and
In conclusion of (i)–(iii), it follows that
On the other hand, the representation formula (4.5) shows that
With a lower bound
The combination of the previously displayed estimates results in
This and (4.13) plus some triangle inequalities imply
Under the overall assumption that
This follows from (A3) for a fixed triangulation
5 Verification of (A4)
The following lemma proves the supercloseness property (1.5) of
Under the overall assumption that
The reduced elliptic regularity of the leading elliptic part
The supercloseness (1.5) is discussed in the introduction and (unlike the remaining results of this paper) holds without any assumption on the initial mesh-size as long as (1.1) is injective and (1.4) has a solution.
The dual problem (5.1) and its solution
The combination of the aforementioned identities in the first step and the identity
The application of these approximation properties to (5.3) concludes the proof. ∎
There exists a constant
This and the Cauchy inequality control the first contribution in (5.4) by
The combination of the two estimates for the two contributions in (5.4) leads (with
This and Corollary 9 with
Axiom (A4) holds for sufficiently small initial mesh-sizes
Several arguments in this section apply to other mixed finite element methods as well but the
second contribution in (5.4) solely applies to the Raviart–Thomas mixed finite element family.
The restriction on the smallness of
6 Numerical Experiments
This section is devoted to numerical experiments to investigate the influence of the critical parameters
6.1 Numerical Realization
The data approximation is realized by the Thresholding Second Algorithm (TSA) of  followed by the closure algorithm to output a shape-regular triangulation.
The realization from [17, 18] is
slightly modified in the Approx algorithm of
 through a parameter
The non-homogeneous boundary data in Section 6.2
are not met in the theoretical part of this paper, which is simplified to
homogeneous boundary conditions. The first example with known solution
requires inhomogeneous boundary data on
The abbreviation error ε (resp. estimator σ) refers to the left-(resp. right-)hand side of (2.1).
6.2 Continuous Right-Hand Side with Known Corner Singularity
in polar coordinates
Figure 1 displays the outcome of Safem (Algorithm 1) with
6.3 Constant Right-Hand Side
The convergence history plot of Figure 3 displays the estimators σ
as functions of the number of degrees of freedom for various initial meshes,
The numerical experiment with the coarsest initial mesh
The undisplayed numerical experiment for a smaller parameter
6.4 Piecewise Constant Right-Hand Side
Given the constant coefficients with
The convergence history plot for this example is not displayed as it looks very similar to that of the previous subsection (namely Figure 3) although the reasons for a larger pre-asymptotic range might be different.
For smaller values of ϵ, the triangulations look more like
the picture in Figure 2 (left) from the previous subsection and
solely Case A runs in Safem. Even for
The overall impression from the displayed and undisplayed numerical experiments is that
the algorithm Safem is very robust such that the choice of
The work has been written, while the first author enjoyed the hospitality of the Hausdorff Research Institute of Mathematics in Bonn, Germany, during the Hausdorff Trimester Program Multiscale Problems: Algorithms, Numerical Analysis and Computation. The authors thank Rui Ma for a careful reading of the manuscript and her valuable remarks.
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