The well-posedness and the a priori and a posteriori error analysis of the lowest-order Raviart–Thomas mixed finite element method (MFEM) has been established for non-selfadjoint indefinite second-order linear elliptic problems recently in an article by Carstensen, Dond, Nataraj and Pani (Numer. Math., 2016). The associated adaptive mesh-refinement strategy faces the difficulty of the flux error control in and so involves a data-approximation error in the norm of the right-hand side f and its piecewise constant approximation . The separate marking strategy has recently been suggested with a split of a Dörfler marking for the remaining error estimator and an optimal data approximation strategy for the appropriate treatment of . The resulting strategy presented in this paper utilizes the abstract algorithm and convergence analysis of Carstensen and Rabus (SINUM, 2017) and generalizes it to general second-order elliptic linear PDEs. The argument for the treatment of the piecewise constant displacement approximation is its supercloseness to the piecewise constant approximation of the exact displacement u. The overall convergence analysis then indeed follows the axioms of adaptivity for separate marking. Some results on mixed and nonconforming finite element approximations on the multiply connected polygonal 2D Lipschitz domain are of general interest.
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 and piecewise constant coefficients in a (possibly multiply connected) bounded polygonal Lipschitz domain , the general second-order linear elliptic PDE seeks such that
The coefficients , , and γ are all piecewise constant functions and the symmetric matrix is assumed to be positive definite with universal positive eigenvalue bounds from below and above. The entire paper solely assumes that is injective (i.e., has a trivial kernel); then it is bijective with a bounded inverse and satisfies a global inf-sup condition. The flux variable with allows to recast problem (1.1) as a first-order system: Seek such that
The mixed formulation of (1.1) seeks such that
for all and for all . The well-posedness of (1.3) has been studied in [4, Theorem 2.1] using the equivalence of the weak formulation of (1.1) and the mixed formulation (1.3). The mixed finite element discretization of (1.3) utilizes the piecewise constant functions on and the lowest-order Raviart–Thomas finite element space and seeks such that
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 and for adaptive conforming FEMs, the optimal convergence rates have been established in  with a collective marking strategy based on a posteriori error estimation, and – in contrast to the results in  – without the interior node property for the refinement. The convergence of adaptive nonconforming FEMs for the non-symmetric and indefinite problem has been derived in  with a different separate marking strategy. The adaptive mixed FEM (1.4) has been considered in  with a combined norm of the norm in the flux error and the norm in the displacement error. Their quasi-optimality analysis is based on some nonstandard adaptive separate marking scheme and a special relation between the mixed FEM and nonconforming schemes.
This paper develops the quasi-optimality of adaptive MFEMs with the natural norm, that is, the combination of a flux error in the norm and the displacement error in the norm via the axioms for separate marking from . The total adaptive estimator is the sum of the residual-based error estimator and the data approximation error . Given a parameter , the separate marking scheme runs either the Dörfler marking  on if or an optimal data approximation algorithm as in [7, 17, 18] to reduce . The main challenge is the proof of the axioms of discrete reliability and quasi-orthogonality for the non-symmetric mixed problem.
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 of the exact displacement u and its supercloseness
in the proof of quasi-orthogonality (A4) below. In fact, this difficulty does not arise in  for . At first glance, the extended Marini-type identity [4, equation (4.3)] states that is close to for some associated Crouzeix–Raviart solution , which is superclose to u by -duality arguments with from reduced elliptic regularity. This argument, however, leads to (1.5) up to an additional term , which is not included in the error estimator η utilized in this paper. In fact, η solely involves the term with a higher power of the localized mesh-size .
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 , , and and their norms with and apply throughout the paper. The notation abbreviates for a mesh-size independent generic constant , which may depend on the domain Ω and the shape but not the size of the triangles of the corresponding shape-regular triangulation. The constant C may also depend on the coefficients , , and γ through lower and upper bounds of the positive eigenvalues of and the upper bound for the remaining coefficients. Furthermore, abbreviates and . The context depending symbol denotes the area of a domain, the length of an edge, the counting measure (cardinality) of a set, the absolute value of a real number, or the Euclidean length of a vector.
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.
Let be an admissible triangulation of the bounded polygonal domain Ω and let be the set of all admissible triangulations refined from by newest vertex bisection . Let denote the set of the three edges of a triangle , let (resp. and ) denote the set of all (resp. interior and boundary) edges in the triangulation and let be the set of its vertices.
Let denote the maximal local mesh-size and let and are the unit normal and tangential vectors along of . The jump of is defined across an interior edge E shared by the two triangles and , which form the edge patch . For any boundary edge , let denote the interior of the triangle with the edge and the jump reduces to the trace (that is, the exterior jump contribution vanishes according to the homogeneous boundary condition). For and , the curl and gradient operators read
The piecewise gradient acts as for all . Let denote the algebraic polynomials of degree at most r and set
Let denote the piecewise projection onto with respect to the shape-regular triangulation . The associated nonconforming Crouzeix–Raviart and lowest-order Raviart–Thomas mixed finite element spaces read
2.2 A Posteriori Error Control
holds for the (squared) error estimator and data approximation defined by
Given the discrete solution and with respect to the admissible triangulation and its refinement , respectively, the distance function reads
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 and and some small positive input parameters and κ.
Algorithm 1 (Safem.).
The routine Refine applies the newest vertex bisection  and refines the marked triangles to compute the smallest admissible refinement of with .
2.4 Axioms and Optimal Convergence
Suppose that there exist universal positive constants and that satisfy (A1)–(A4) and (B1)–(B2) below. Here and in the following is an admissible triangulation and refinement of some . Recall the definition of the error estimator in (2.2)–(2.4) and the abbreviation for all .
Stability: For all and all ,
Reduction: For all and all ,
Discrete Reliability: For all and all ,
Let and be the output of Safem of Section 2.3 and abbreviate and , etc.
Quasi-Orthogonality: For all ,
Rate s Data Approximation: There exists such that for , satisfies
Quasimonotonicity of μ: For all and all ,
Theorem 1 (Quasi-Optimality ).
The proof of Theorem 1 is given in  and not recalled here. The version of this paper is even slightly simplified in that , , and are more general in  and are not displayed explicitly in this paper.
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 are bounded by the constant in the discrete jump control.
Lemma 1 (Discrete Jump Control ).
There exists a universal constant , which depends on the interior angles in the regular triangulation and the degree , such that any with jumps
The discrete jump control plus triangle inequalities in Lebesgue spaces and in finite-dimensional Euclidean spaces imply the stability (A1). Throughout this section, let and denote the discrete solution with respect to and its refinement , respectively, and let be the distance function (2.4).
Theorem 2 ((A1) Stability).
Axiom (A1) holds with for a global lower bound of the piecewise constant eigenvalues of the coefficient matrix , the supremum of , the maximal mesh-size , and for the constant from the discrete jump control of Lemma 1.
The reverse triangle inequality in for over the element contributions implies that
Each of the terms and is a norm in of terms, which are Lebesgue norms and so allow for a reverse triangle inequality. This leads to
with the abbreviation . The sum over all involves volume terms and edge jumps. Lemma 1 controls the latter terms and so results in
The mesh-size is bounded from above and so the right-hand side is bounded by the factor times the squared norm of g. A triangle inequality and the bounds on the coefficients show
For the m refined triangles of , the sum reads
The reverse triangle inequalities in and in Lebesgue spaces over triangles and edges and the abbreviation g from the previous proof plus show
Since for and for , the first term on the right-hand side of the above displayed formula is bounded from above by . The remaining term is estimated with Lemma 1 as in the previous proof and so with the same bound on . ∎
4 Verification of (A3)
Throughout this section let and solve (1.4) and let and denote the orthogonal projection of the right-hand side f onto piecewise constants ( and ) with respect to the triangulation and its refinement , respectively.
The main residual is defined, for any test function , by
There exists some with norm and
The initial mesh-size is sufficiently small throughout this paper to guarantee the existence and stability of the discrete solutions [4, Theorem 4.3]. The stability of the discrete problem (1.4) with respect to the refined triangulation leads to the existence of with and
A Cauchy inequality concludes the proof. ∎
The further analysis of requires a discrete Helmholtz decomposition on a regular triangulation of a (possibly) multi-connected domain Ω. The connectivity components of are enumerated such that denotes the boundary of the unbounded component of The modified lowest-order Crouzeix–Raviart space reads
(Here and throughout the paper, denotes the set of edges on .)
Lemma 2 (Discrete Helmholtz Decomposition).
For the multi-connected domain Ω the decomposition of piecewise constant vector functions
is orthogonal with respect to the scalar product weighted by in the sense that
The discrete Helmholtz decomposition is well known for simply-connected domains and . The general case follows with the same argument by counting triangles and edges in general; further details are omitted. ∎
The modified Crouzeix–Raviart space is accompanied by a modified Raviart–Thomas space with vanishing integral mean of the normal components over each boundary,
The discrete Helmholtz decomposition allows for a characterization of the divergence-free functions.
Lemma 3 (Discrete Divergence).
The linear operator
is surjective and its kernel is .
The divergence-free Raviart–Thomas functions in are piecewise constant and allow for a discrete Helmholtz decomposition as in Lemma 2. The decomposition implies that the divergence-free Raviart–Thomas functions in are those in . Consequently, the kernel of has the dimension for the number of nodes in . Since the dimension of the vector space is for the number of edges in , the range of has dimension . Since , the range is . ∎
One key argument of the reliability analysis is the split of the difference into two parts and for some auxiliary solution: Seek with
hold for all , for all , and for all .
The solution to (4.3) is recovered from an auxiliary nonconforming problem: Let denote the Riesz representation of the functional on the right-hand side of
for all test functions in the Hilbert space . Here and throughout the paper, denotes the integral mean for the length of the closed polygon .
The piecewise constant function is
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 present situation involves the connectivity components of the boundary and requires a little modification in the proof that indeed satisfies (4.3). Set
and substitute from (4.5) before a piecewise integration by parts shows, for any test function , that
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 in the aforementioned equality and leads with (4.6) to . Recall that and that for because of . This calculation leads to equation (4.3b). Since , the definition of (4.5) immediately proves (4.3c). Rewrite (4.5) to obtain an identity for and utilize this in (4.4). This leads to an identity, which allows a piecewise integration by parts and then results in
The design of with piecewise integral means along the boundary edges of which are constant for each proves (4.3a).
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
for all , plus for . Given an arbitrary solution to this discrete homogeneous problem, let to find . It follows for all test functions. Lemma 3 allows for a test function with and so . This concludes the proof of the uniqueness of the solution of the homogeneous system. ∎
The test function in Lemma 1 can be selected additionally to satisfy and for all .
The first equation in (1.4) shows that belongs to the kernel of from (4.1). Hence we may and will replace test function in Lemma 1 by for some appropriate . The naive choice of the Fortin interpolation of leads to the additional properties but leaves open the subtle question of the uniform bound in terms of . This proof utilizes the MFEM solution to the modified Poisson model problem with right-hand side ,
Lemma 5 and the modified test-function show
The divergence-free term is controlled in the subsequent lemma.
It holds .
Since implies , it follows that is piecewise constant and its discrete Helmholtz decomposition leads to and with
This and the orthogonality show that . Consequently,
Given any node in the coarse triangulation , the Scott–Zhang quasi-interpolation  defines by a selection of an edge with vertex z and evaluates some weighted integral of along . Select the edge if possible to obtain a Scott–Zhang quasi-interpolation of with a.e. in plus the local approximation and stability properties. For any edge of length and its neighborhood for the nodal patches , the latter properties and discrete trace inequalities result in
The piecewise of the low-order Raviart–Thomas finite element functions vanishes and so do all summands in the last term. Since along any edge , this proves
The combination with estimate (4.11) for and the bound imply
A rearrangement with the triangle-oriented error estimator concludes the proof. ∎
Let abbreviate the function for and consider the test function from Lemma 5. The piecewise constant part allows a discrete Helmholtz decomposition
from Lemma 2 for some and some . Altogether, there are three contributions of
(i) The representation of in (4.5) is utilized in the first contribution and leads to
On the other hand, and is orthogonal to . This and an integration by parts prove
Recall that is orthogonal onto and so
Note that vanishes a.e. in and a piecewise discrete Poincaré inequality shows with the interior of the domain . Consequently,
(ii) Observe from (4.3b) and that .
(iii) Since vanishes outside the set (for on ) and the weight satisfies , the term is equal to
In conclusion of (i)–(iii), it follows that
On the other hand, the representation formula (4.5) shows that
With a lower bound of the smallest eigenvalue of A and with from (1.4),
For each , and an inverse estimate plus a uniform upper bound of the eigenvalues of lead to
The combination of the previously displayed estimates results in
This and (4.13) plus some triangle inequalities imply
Since , this concludes the proof. ∎
Theorem 8 ((A3) Discrete Reliability).
Under the overall assumption that is sufficiently small, there exists some universal constant , which depends on the global lower and upper bounds of the eigenvalues of and on the universal stability constant of the discrete problems and on the shape-regularity in such that the following holds. The discrete solutions and of (1.4) with respect to the triangulation and its refinement satisfy
This follows from (A3) for a fixed triangulation and a sequence of its successive uniform refinements as then the maximal mesh-size in tends to zero and standard estimates show convergence of to in the norm of . ∎
5 Verification of (A4)
The following lemma proves the supercloseness property (1.5) of to the mixed solution with a duality argument. For given right-hand side , the dual problem seeks with
Under the overall assumption that is injective, it follows that and its dual are isomorphisms between and .
The reduced elliptic regularity of the leading elliptic part leads to higher regularity, that is, there exist α with and with
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 for the right-hand side lead to for some . This allows the application of the Fortin interpolation operator [3, pp. 107–109] with the commutative property . This and result in
The combination of the aforementioned identities in the first step and the identity in the second step plus an integration by parts prove that
The application of these approximation properties to (5.3) concludes the proof. ∎
Recall that is the positive extra regularity parameter, which exclusively depends on the domain and on the coefficients, and the maximal initial mesh-size is the maximal mesh-size in (whence in all ) and is from (A3).
Theorem 2 ((A) Quasi-Orthogonality).
There exists a constant such that for sufficiently small , any satisfy
Recall and abbreviate
The factor in this scalar product is split into and with the projection onto . Lemma 1 applies on the level of and controls the norm
This and the Cauchy inequality control the first contribution in (5.4) by times the constant . The remaining second contribution in term (5.4) is the scalar product of with (recall that is a constant vector with length ). The Raviart–Thomas function allows in 2D for
at and so
The combination of the two estimates for the two contributions in (5.4) leads (with for and ) to
Provided that , the sum of the above inequalities over different levels shows
This and Corollary 9 with conclude the proof. ∎
Remark 4 (Generalizations).
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 in Theorem 2 can be circumvented by the quasi-monotonicity (cf.  for the concept) but is required in Corollary 3.
6 Numerical Experiments
This section is devoted to numerical experiments to investigate the influence of the critical parameters , , and κ and the practical performance of the adaptive algorithm Safem. After a few remarks on the implementation, three examples on the L-shaped domain are displayed with smooth or discontinuous right-hand sides, before some overall observations conclude the paper.
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.
Algorithm 2 (TSA.).
The realization from [17, 18] is slightly modified in the Approx algorithm of  through a parameter in the computation of . The functional in TSA is a weighted error functional, which depend on the values of and on the parent triangle as well as on the siblings of , cf. [1, 17, 18] for more details and the explicit formulas.
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 with the modified jump-term along the boundary edge with the prescribed boundary values u and its tangential derivative on E.
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 centered at the origin. The initial mesh consists of 24 congruent right-isosceles triangles from a criss refinement of the three sub-squares and 28 degrees of freedom.
Figure 1 displays the outcome of Safem (Algorithm 1) with , , and various values of , 0.3, and 0.5. Despite the condition of Theorem 1 on the smallness of the bulk parameter , all displayed values result in an improved optimal empirical convergence rate compared to uniform mesh-refinement with a known suboptimal convergence rate . The initial mesh is relatively coarse but all convergence rates are visible right from the beginning, the pre-asymptotic range is not visible. This examples allows the computation of the errors and the estimators and the equivalence is visible throughout with a the expected behavior. In the displayed experiment with only Case A of Safem applies as the right-hand side f is continuous. In case of for instance, only Case B runs in Safem for a very long computational range.
6.3 Constant Right-Hand Side
The coefficients , and on the L-shaped domain Ω with constant right-hand side lead in (1.1) to an unknown weak solution . Figure 2 (left) displays the output of through Safem with and . Besides the local mesh-refining at the re-entering corner, some layer of refinement are visible along some part of the boundary, that mimics a singular perturbed situation with for very small ϵ. Undisplayed numerical experiments for have confirmed this observation even stronger.
The convergence history plot of Figure 3 displays the estimators σ as functions of the number of degrees of freedom for various initial meshes, namely for as described in the previous subsection and also for an initial mesh (of from the previous subsection) with ; red-refinement means the division of each triangle into four congruent sub-triangles by connecting its edges’ midpoints with straight lines. This is plotted under the label σ(uniform) and shows the expected suboptimal empirical convergence rate. Those red-refined triangulations, e.g., (with ) with two red-refinements and (with ) for three, serve as initial triangulations in the input of Safem and Figure 3 displays the respective convergence history plots.
The numerical experiment with the coarsest initial mesh displays a pre-asymptotic range up to 1000 degrees of freedom. The finer initial triangulations lead to a much smaller pre-asymptotic range with a rapid decrease through a strong local mesh-refinement until the convergence rate of the other adaptive mesh-refinements is met. For the displayed parameter , solely the Case A runs in Safem.
The undisplayed numerical experiment for a smaller parameter leads to a much larger pre-asymptotic domain with a systematic error reduction only for a fine initial mesh .
6.4 Piecewise Constant Right-Hand Side
Given the constant coefficients with and the domain as in the previous subsection, the right-hand side f for this example is piecewise constant with the values and the value -1 exactly on the two squares not aligned to the triangulations; and . Figure 2 (right) displays the output of Safem with and with two squares ω visible by local mesh-refinements along to resolve the discontinuity of the right-hand side f (recall that f is discontinuous at triangles that intersect ). Cases A and B alternate in Safem for this example with for the resolution of the discontinuities of the right-hand side at .
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 (and more so for ), the mesh-refinement displays boundary layers and no longer the discontinuities along .
The overall impression from the displayed and undisplayed numerical experiments is that the algorithm Safem is very robust such that the choice of , , κ in the asymptotic convergence regime with an observed optimal convergence rate: The values and can be recommended throughout the examples of this paper. The condition on a sufficiently fine initial mesh dramatically influences the pre-asymptotic behavior. Although the examples in Subsection 6.3 and 6.4 are very different in the right-hand side, the stability of the discrete system is identical. The finer the initial mesh, the smaller is the pre-asymptotic range in particular for with very small ϵ (e.g., ). This paper exploits the situation when solely is injective and then appears necessary for the well-posedness of the discrete systems and has to be monitored in practise. It is conjectured that this dominates the difficulty of choosing an appropriate initial triangulation in Safem.
Funding statement: The research has been supported by the Deutsche Forschungsgemeinschaft in the Priority Program 1748 under the project Foundation and application of generalized mixed FEM towards nonlinear problems in solid mechanics (CA 151/22-1). The research was carried out while the second author (Asha K. Dond) enjoyed the support of the Priority Program 1748 Reliable simulation techniques in solid mechanics. Development of non-standard discretization methods, mechanical and mathematical analysis for a research visit at the Humboldt-Universität zu Berlin, Germany.
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.
 P. Binev, W. Dahmen and R. DeVore, Adaptive finite element methods with convergence rates, Numer. Math. 97 (2004), no. 2, 219–268. Search in Google Scholar
 P. Binev and R. DeVore, Fast computation in adaptive tree approximation, Numer. Math. 97 (2004), no. 2, 193–217. Search in Google Scholar
 D. Boffi, F. Brezzi and M. Fortin, Mixed Finite Element Methods and Applications, Springer Ser. Comput. Math. 44, Springer, Heidelberg, 2013. Search in Google Scholar
 C. Carstensen, A. K. Dond, N. Nataraj and A. K. Pani, Error analysis of nonconforming and mixed FEMs for second-order linear non-selfadjoint and indefinite elliptic problems, Numer. Math. 133 (2016), no. 3, 557–597. Search in Google Scholar
 C. Carstensen, M. Feischl, M. Page and D. Praetorius, Axioms of adaptivity, Comput. Math. Appl. 67 (2014), no. 6, 1195–1253. Search in Google Scholar
 C. Carstensen and R. H. W. Hoppe, Error reduction and convergence for an adaptive mixed finite element method, Math. Comp. 75 (2006), no. 255, 1033–1042. Search in Google Scholar
 C. Carstensen and H. Rabus, An optimal adaptive mixed finite element method, Math. Comp. 80 (2011), no. 274, 649–667. Search in Google Scholar
 C. Carstensen and H. Rabus, Axioms of adaptivity with separate marking for data resolution, SIAM J. Numer. Anal. 55 (2017), no. 6, 2644–2665. Search in Google Scholar
 J. M. Cascon, C. Kreuzer, R. H. Nochetto and K. G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal. 46 (2008), no. 5, 2524–2550. Search in Google Scholar
 H. Chen, X. Xu and R. H. W. Hoppe, Convergence and quasi-optimality of adaptive nonconforming finite element methods for some nonsymmetric and indefinite problems, Numer. Math. 116 (2010), no. 3, 383–419. Search in Google Scholar
 L. Chen, M. Holst and J. Xu, Convergence and optimality of adaptive mixed finite element methods, Math. Comp. 78 (2009), no. 265, 35–53. Search in Google Scholar
 M. Dauge, Elliptic Boundary Value Problems on Corner Domains. Smoothness and Asymptotics of Solutions, Lecture Notes in Math. 1341, Springer, Berlin, 1988. Search in Google Scholar
 A. K. Dond, N. Nataraj and A. K. Pani, Convergence of an adaptive lowest-order Raviart–Thomas element method for general second-order linear elliptic problems, IMA J. Numer. Anal. 37 (2017), no. 2, 832–860. Search in Google Scholar
 W. Dörfler, A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal. 33 (1996), no. 3, 1106–1124. Search in Google Scholar
 M. Feischl, T. Führer and D. Praetorius, Adaptive FEM with optimal convergence rates for a certain class of nonsymmetric and possibly nonlinear problems, SIAM J. Numer. Anal. 52 (2014), no. 2, 601–625. Search in Google Scholar
 K. Mekchay and R. H. Nochetto, Convergence of adaptive finite element methods for general second order linear elliptic PDEs, SIAM J. Numer. Anal. 43 (2005), no. 5, 1803–1827. Search in Google Scholar
 H. Rabus, Quasi-optimal convergence of AFEM based on separate marking, Part I, J. Numer. Math. 23 (2015), no. 2, 137–156. Search in Google Scholar
 H. Rabus, Quasi-optimal convergence of AFEM based on separate marking, Part II, J. Numer. Math. 23 (2015), no. 2, 157–174. Search in Google Scholar
 L. R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), no. 190, 483–493. Search in Google Scholar
 R. Stevenson, Optimality of a standard adaptive finite element method, Found. Comput. Math. 7 (2007), no. 2, 245–269. Search in Google Scholar
 R. Stevenson, The completion of locally refined simplicial partitions created by bisection, Math. Comp. 77 (2008), no. 261, 227–241. Search in Google Scholar
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