Non-conforming finite element methods (FEMs) play an important role in computational mechanics. They allow the discretization of partial differential equations (PDEs) for incompressible fluid flows, for almost incompressible materials in linear elasticity, and for low polynomial degrees in the ansatz spaces for higher-order problems. The projection property of the interpolation operator of the non-conforming FEM, also named after Crouzeix and Raviart , states that the projection of onto the space of piecewise constant functions equals the space of piecewise gradients of the non-conforming interpolation of functions in the non-conforming finite element space. This property is the basis for the proof of the discrete inf-sup condition for the Stokes equations  as well as for the analysis of adaptive algorithms .
Many possible generalizations of the non-conforming FEM to higher polynomial degrees have been proposed. All those generalizations are either based on a modification of the classical concept of degrees of freedom [23, 22, 41], are restricted to odd polynomial degrees [20, 3], or employ an enrichment by additional bubble functions [29, 28]. However, none of those generalizations possesses a corresponding projection property of the interpolation operator for higher moments (see Remark 3.15 below). This paper introduces a novel formulation of the Poisson equation (in (3.3) below) based on the Helmholtz decomposition along with its discretization of arbitrary (globally fixed) polynomial degree. This new discretization approximates directly the gradient of the solution, which is often the quantity of interest, instead of the solution itself. For the lowest-order polynomial degree, the discrete Helmholtz decomposition of  proves equivalence of the novel discretization with the known non-conforming Crouzeix–Raviart FEM  and therefore they appear in a natural hierarchy. In the context of the novel (mixed) formulation, these discretizations turn out to be conforming.
Although the complexity of the new discretization itself is competitive with that of a standard FEM, the method requires the pre-computation of some function φ such that its divergence equals the right-hand side. If this is not computable analytically, this results in an additional integration (see also Remark 3.6 below). However, this paper focuses on the Poisson problem as a model problem to introduce the idea of the new approach and to give a broad impression over possible extensions as quadrilateral discretizations (including a discrete Helmholtz decomposition on quadrilateral meshes for the non-conforming Rannacher–Turek FEM  as a further highlight of this paper), the generalization to three dimensions, or inhomogeneous mixed boundary conditions. The advantages of the new approach in the applications of linear elasticity, the Stokes equation, and higher-order problems (including the Kirchhoff plate) are the topic of the papers [36, 37].
The presence of singularities for non-convex domains usually yields the same sub-optimal convergence rate for any polynomial degree. This motivates adaptive mesh-generation strategies, which recover the optimal convergence rates. This paper presents an adaptive algorithm and proves its optimal convergence. The proof essentially follows ideas from the context of the non-conforming Crouzeix–Raviart FEM [6, 32]. This illustrates that the novel discretization generalizes it in a natural way. Since the efficient and reliable error estimator involves a data approximation term without a multiplicative power of the mesh-size, the adaptive algorithm is based on separate marking.
A possible drawback of the new FEMs is that the gradient of the solution is approximated, but not the solution u itself. This excludes obvious generalizations to partial differential equations where u appears in lower-order terms.
The remaining parts of this paper are organized as follows. Section 2 defines some notation. Section 3 introduces the novel formulation based on the Helmholtz decomposition and its discretization together with an a priori error estimate. The equivalence with the non-conforming FEM for the lowest-order case is proved in Section 3.3. Section 4 summarizes some generalizations. Section 5 is devoted to a medius analysis of the FEM, which uses a posteriori techniques to derive a priori error estimates. Section 6 proves quasi-optimality of an adaptive algorithm, while Section 7 outlines the generalization to 3D. Section 8 concludes this paper with numerical experiments.
Throughout this paper is a simply connected, bounded, polygonal Lipschitz domain. Standard notation on Lebesgue and Sobolev spaces and their norms is employed with scalar product . Given a Hilbert space X, let resp. denote the space of functions with values in X whose components are in resp. , and let denote the subset of of functions with vanishing integral mean. The space of functions whose weak divergence exists and is in is denoted with . The space of infinitely differentiable functions reads and the subspace of functions with compact support in Ω is denoted with . The piecewise action of differential operators is denoted with a subscript . The formula represents an inequality for some mesh-size independent, positive generic constant C; abbreviates . By convention, all generic constants do neither depend on the mesh-size nor on the level of a triangulation but may depend on the fixed coarse triangulation and its interior angles. The Curl operator in two dimensions is defined by for sufficiently smooth β.
A shape-regular triangulation of a bounded, polygonal, open Lipschitz domain is a set of closed triangles such that and any two distinct triangles are either disjoint or share exactly one common edge or one vertex. Let denote the edges of a triangle T and the set of edges in . Any edge is associated with a fixed orientation of the unit normal on E (and denotes the unit tangent on E). On the boundary, is the outer unit normal of Ω, while for interior edges , the orientation is fixed through the choice of the triangles and with and is then the outer normal of on E. In this situation, denotes the jump across E. For an edge on the boundary, the jump across E reads . For and , let denote the set of polynomials on a triangle T, and the set of piecewise polynomials, i.e.,
and let . Given a subspace , let denote the projection onto X and let abbreviate . Given a triangle , let denote the square root of the area of T and let denote the piecewise constant mesh-size with for all . For a set of triangles , let abbreviate
Given an initial triangulation , an admissible triangulation is a regular triangulation which can be created from by newest-vertex bisection . The set of admissible triangulations is denoted by .
3 Problem Formulation and Discretization
This section introduces the new formulation based on the Helmholtz decomposition in Section 3.1 and its discretization in Section 3.2. Section 3.3 discusses the equivalence with the non-conforming Crouzeix–Raviart FEM .
3.1 New Mixed Formulation of the Poisson Problem
Given the simply connected, bounded, polygonal Lipschitz domain and , the Poisson model problem seeks with
The novel weak formulation is based on the classical Helmholtz decomposition 
for any simply connected domain , where the sum is orthogonal with respect to the scalar product.
Note that for , the definition of the Curl implies
Define and and let satisfy . The novel weak formulation of the Poisson problem (3.1) seeks with
This formulation is the point of departure for the numerical approximation of in Section 3.2.
(Existence of Solutions)
Since , any satisfies the inf-sup condition
and, hence, solves (3.1).
(Mixed Boundary Conditions)
Let with closed, , and each connectivity component of has positive length. Assume that the triangulation resolves . Let denote the space of generalized normal traces of functions and let and in the sense that there holds on in the sense of distributions for some . Consider the mixed boundary value problem in Ω with on and on . Let denote the subspace of of functions with vanishing trace on . For , define . Define
The Helmholtz decomposition
for mixed boundary conditions [24, Corollary 3.1] then leads to the following formulation. Let with additionally fulfil the boundary condition and seek with
Since , the equivalence follows as in Section 3.1 and with
(Multiply Connected Domains)
If is a multiply connected polygonal bounded Lipschitz domain and , such that all parts of lie on the outer boundary of Ω (on the unbounded connectivity component of ), then the Helmholtz decomposition of Remark 3.4 still holds and a discretization as above is then immediate. However, if the Dirichlet boundary also covers parts of interior boundary, that Helmholtz decomposition does no longer hold: There exist harmonic functions which are constant on different parts of and, hence, are neither in , nor in .
(Computation of φ)
The computation of φ appears as a practical difficulty because φ needs to be defined through an integration of f. If f has some simple structure, e.g., f is polynomial, this can be done manually, while for more complicated f, can be defined by a numerical integration of f, e.g., by approximating
for some with and defined by and . This is possible in parallel.
Let be a regular triangulation of Ω and and define
The discretization of (3.3) seeks and with
Since there are no continuity conditions on and since , equation (3.4a) is fulfilled in a strong form, i.e.,
In contrast to classical finite element methods, the approximation of is a gradient only in a discrete orthogonal sense, namely (3.4b). For , Section 3.3 below proves that this discrete orthogonal gradient property is equivalent to being a non-conforming gradient of a Crouzeix–Raviart finite element function. The main motivation of the novel formulation is the generalization of this scheme to any polynomial degree k.
(Existence of Discrete Solutions)
Since , the discrete inf-sup condition
follows from the orthogonality of and .
A direct analysis of the bilinear form defined by
for all and all reveals that the inf-sup constant of equals 5 and, hence, the constant hidden in in (3.5) is 5.
The best-approximation of Theorem 3.9 contains the term
on the right-hand side, which depends on the choice of φ. This seems to be worse than the best-approximation results for standard FEMs, which do not involve such a term. However, if φ is chosen smooth enough, then has at least the same regularity as , and therefore the convergence rate is not diminished. On the other hand, the approximation space for p does not have any continuity restriction and so the first approximation term
is superior to the best-approximation of a standard FEM, where is approximated with gradients of finite element functions. However, [42, Theorem 3.2] and the comparison results of  prove equivalence of (3.6) and the best-approximation with gradients of a standard FEM up to some multiplicative constant.
The following lemma proves a projection property. This means that for any , the best-approximation of in is a discrete orthogonal gradient in the sense that it is orthogonal to and so belongs to the set of discrete orthogonal gradients defined by
The projection property is the key ingredient in the optimality analysis of Section 6.
It holds that . Moreover, if is an admissible refinement of , then .
Let . Since and , the orthogonality in the Helmholtz decomposition (3.2) implies for any that
This proves . For the converse direction, let and let be a solution (possibly not unique) to
The orthogonality of to implies the existence of such that . Therefore, and, hence, α is a piecewise polynomial of degree and therefore . But since , it holds that
and, hence, . This proves and, therefore, .
A similar proof applies in the discrete case and proves . ∎
Problem (3.4) is equivalent to: Find such that
Therefore, the system matrix is (in 2D) the same as that of a standard FEM (up to degrees of freedom on the boundary).
3.3 Equivalence with Crouzeix–Raviart FEM
The non-conforming Crouzeix–Raviart finite element space  reads
Since (if the triangulation consists of more than one triangle), the weak gradient of a function does not exist in general. However, the piecewise version defined by for all exists. The non-conforming discretization of the Poisson problem seeks with
The lowest-order space of Raviart–Thomas finite element functions  reads
The Raviart–Thomas functions have the property that the integration by parts formula holds for functions in as well as for functions in .
The following proposition proves the equivalence of the non-conforming discretization and the discretization (3.4) for . Note that the discretization (3.8) is a non-conforming discretization, while the discretization (3.4) is a conforming one.
(Equivalence with CR-NCFEM)
The crucial point is the discrete Helmholtz decomposition 
Since is orthogonal to , this implies for some . Let for some . Then is orthogonal to and a piecewise integration by parts and (3.4) imply
Hence, solves (3.8). ∎
The projection property from Lemma 3.12 generalizes the famous integral mean property
of the non-conforming interpolation operator .
(Higher Polynomial Degrees)
For higher polynomial degrees , the discretization (3.4) is not equivalent to known non-conforming schemes [23, 20, 21, 28], in the sense that for those non-conforming finite element spaces . This follows from for non-conforming FEMs with enrichment. A dimension argument shows
for the non-conforming FEMs of [23, 20] without enrichment and therefore . Moreover, this proves that the generalization of the projection property to higher moments from Lemma 3.12 cannot hold for those finite element spaces, in contrast to the discretization (3.4).
Section 4.1 generalizes the novel FEM to quadrilateral meshes and proves a new discrete Helmholtz decomposition for the rotated non-conforming Rannacher–Turek FEM . Section 4.2 discusses a discretization with Raviart–Thomas functions.
4.1 Quadrilateral Finite Elements
For this subsection, consider a regular partition of Ω in quadrilaterals. Define, for the reference rectangle ,
Given , let denote the bilinear transformation from the reference rectangle to T. For consistency, let and set
Then a discretization with respect to the quadrilateral partition seeks and with
Let , i.e., . A direct calculation reveals, for all ,
Let with and . Then it holds
This implies and . The combination of the previous equalities leads to
Consequently, . This and the conformity of the method prove as in Section 3 the following statements:
unique existence of solutions,
the best-approximation result
the projection property
The properties (i)–(iii) still hold for any with .
The remaining part of this subsection proves the equivalence of the lowest-order rectangular discretization with the non-conforming Rannacher–Turek FEM . To this end, define for the reference rectangle and the bilinear transformation ,
The following lemma proves a relation between the cardinalities of the quadrilaterals, nodes, and interior edges of a quadrilateral partition similar to Euler’s formulae
on triangles. This enables a dimension argument in the proof of the discrete Helmholtz decomposition in Theorem 4.3 below.
(Euler Formula for Quadrilateral Partitions)
Let be a regular partition of Ω in quadrilaterals with edges , interior edges , and vertices . Then it holds that .
Define a triangulation of Ω in triangles by the division of each quadrilateral into two triangles by a diagonal cut. Let denote the edges of , the interior edges and the vertices. Then the following relations between the two partitions hold:
This and Euler’s formulae for triangles (4.3) prove
The following theorem proves that the solution space from (4.1) equals the piecewise gradients of functions in on a partition in squares for .
(Discrete Helmholtz Decomposition on Squares)
Let be a regular partition of Ω in squares. Then,
and the decomposition is orthogonal.
The orthogonality in (4.4) still holds for a partition in parallelograms. However,
for general quadrilateral partitions.
Proof of Theorem 4.3.
Let and . A piecewise integration by parts leads to
Since consists of parallelograms, the bilinear transformation is affine and, hence, is affine on each edge . This implies that is constant. Since the integral mean of vanishes, this proves the orthogonality.
Let . A computation reveals for all that there exist and such that
For , reads
Since all are squares, and commute, and, hence, . Thus,
The dimension of equals and the dimension of equals , while the dimension of equals . This and Lemma 4.2 prove the assertion. ∎
The best-approximation (ii) from above proves quasi-optimal convergence even for arbitrary quadrilaterals. Standard interpolation error estimates for and for (see ) lead to first-order convergence rates of h for sufficiently smooth solutions. This should be contrasted with , where quasi-optimal convergence is only obtained for a modification of (4.2) where is defined in terms of local coordinates.
4.2 Relation to Mixed Raviart–Thomas FEM
This subsection shows that the classical mixed Raviart–Thomas FEM  can be regarded as a particular choice of the ansatz spaces in the new mixed scheme.
Let denote a regular triangulation of Ω in triangles. Define the space of Raviart–Thomas functions 
Then the following problem is a discretization of (3.3): Find with
with the operator defined for all by
This decomposition yields the equivalence of (4.5) with the problem: Find with
This is the classical Raviart–Thomas discretization with f replaced by .
Assume now that the right-hand side is a Raviart–Thomas function. Since by definition with from Section 3.2 and since is the solution of
it holds with from (3.4). Since and , it follows
For , the equivalence with the Crouzeix–Raviart FEM (3.9) then proves the identity
5 Medius Analysis
The following theorem proves a generalization for the discretization (3.4) for the lowest order case .
If φ is a lowest-order Raviart–Thomas function, then it allows for an integration by parts formula also with Crouzeix–Raviart functions (see Section 3.3). Therefore, the third term on the right-hand side of (5.2) vanishes. This and the equivalence with the non-conforming FEM of Crouzeix and Raviart from Section 3.3 reveal the best-approximation result (5.1).
For any there exists with the following properties:
Proof of Theorem 5.1.
For the first term on the right-hand side, properties (i) and (iii) from Lemma 5.3 yield
Since , it follows
Properties (ii) and (iii) of Lemma 5.3 prove
prove the assertion. ∎
6 Adaptive Algorithm
This section defines an adaptive algorithm based on separate marking and proves its quasi-optimal convergence.
6.1 Adaptive Algorithm and Optimal Convergence Rates
Let denote some initial shape-regular triangulation of Ω, such that each triangle is equipped with a refinement edge . A proper choice of these refinement edges guarantees an overhead control .
Let denote the subset of of all admissible triangulations with at most triangles. The adaptive algorithm involves the overlay of two admissible triangulations , which reads
Given a triangulation , define for all the local error estimator contributions by
and the global error estimators by
for any . The adaptive algorithm is driven by these two error estimators and runs the following loop.
(Separate Versus Collective Marking)
The residual-based error estimator involves the term without a multiplicative positive power of the mesh-size. Therefore, the optimality of an adaptive algorithm based on collective marking (that is and λ replaced by in Algorithm 1) does not follow from the abstract framework from . The reduction property (axiom (A2) from ), is not fulfilled. Algorithm 1 considered here is based on separate marking. In this context, the optimality of the adaptive algorithm (see Theorem 6.5) can be proved with a reduction property that only considers λ.
The step Mark in the second case () can be realized by the algorithm Approx from [7, 16], i.e., the thresholding second algorithm  followed by a completion algorithm. For this algorithm, the assumption (B1) optimal data approximation, which is assumed to hold in the following, follows from the axioms (B2) and (SA) from Section 6.5, cf. . For a discussion about other algorithms that realize Mark in the second case, see again .
For and define
(Pure Local Approximation Class)
Since Ω is assumed to be a Lipschitz domain, all patches in an admissible triangulation are edge-connected, i.e., for all vertices and triangles with , there exists and with , , and for all . Under this assumption, [42, Theorem 3.2] shows
In the following, we assume that the following assumption (B1) holds for the algorithm used in the step Mark for (see Remark 6.2).
((B1) Optimal Data Approximation)
Assume that is finite. Given a tolerance , the algorithm used in Mark in the second case () in Algorithm 1 computes with
The following theorem states optimal convergence rates of Algorithm 1.
(Optimal Convergence Rates of AFEM)
For and sufficiently small and , Algorithm 1 computes sequences of triangulations and discrete solutions for the right-hand side φ of optimal rate of convergence in the sense that
The proof follows from the abstract framework of , which employs the bounded overhead  of the newest-vertex bisection, under the assumptions (A1)–(A4) and (B2) and (SA) which are proved in Sections 6.2–6.5.
6.2 (A1) Stability and (A2) Reduction
The following two theorems follow from the structure of λ.
Let be an admissible refinement of and . Let and be the respective discrete solutions to (3.4). Then,
Let be an admissible refinement of . Then there exist and such that
This follows with a triangle inequality and the mesh-size reduction property for all as in [18, Corollary 3.4]. ∎
6.3 (A4) Discrete Reliability
The following theorem proves discrete reliability, i.e., the difference between two discrete solutions is bounded by the error estimators on refined triangles only.
Let be an admissible refinement of with respective discrete solutions and . Then,
Recall the definition of from (3.7). Since , there exist and with . Since , it holds
The orthogonality furthermore implies that the discrete error can be split as
For any triangle , it holds . Therefore,
Since is a refinement of , it holds
Let denote the quasi interpolant from  of which satisfies the approximation and stability properties
and for all edges . Since and , it holds
An integration by parts leads to
For a triangle , any edge satisfies . Hence, for all triangles . This, the Cauchy inequality and the approximation and stability properties of the quasi interpolant lead to
Since for all edges , the approximation and stability properties of the quasi interpolant and the trace inequality [10, p. 282] lead to
The combination of the previous displayed inequalities yields
Since and , the triangle inequality yields the assertion. ∎
The discrete reliability of Theorem 6.8 together with the convergence of the discretization proves reliability of the residual-based error estimator. This is summarized in the following proposition.
(Efficiency and Reliability of the Residual-Based Error Estimator)
6.4 (A3) Quasi-Orthogonality
The following theorem proves quasi-orthogonality of the discretization (3.4).
Let be some sequence of triangulations with discrete solutions to (3.4). Let . Then,
The subtraction of these two equations and an index shift lead, for any with , to
Since is orthogonal to , a Cauchy and a weighted Young inequality imply
The orthogonality for all proves
The definition of yields
This and a further application of Theorem 6.8 lead to
The Young inequality, the triangle inequality, and imply
7 Extension to 3D
This section is devoted to the generalization to 3D. Section 7.1 defines the novel discretization and comments on basic properties, while Section 7.2 is devoted to optimal convergence rates for the adaptive algorithm.
7.1 Weak Formulation and Discretization
For this section, let be a simply connected, bounded, polygonal Lipschitz domain in . For the sake of simplicity, we also assume that is connected (i.e., Ω is contractible). The Curl operator acts on a sufficiently smooth vector field as with the cross product or vector product . Let denote the space of all with for the weak , i.e.,
In contrast to the two-dimensional case, . The Helmholtz decomposition in 3D reads
and the sum is orthogonal. It is a consequence of the identity
in the De Rham complex .
Let with . Then the Poisson problem (3.1) is equivalent to the problem: Find with
In contrast to the two-dimensional case, the operator has a non-trivial kernel. Classical results  characterize this kernel as . To enforce uniqueness, we can reformulate (7.1) as follows: Find with
Note that implies .
Standard finite element spaces to discretize in 3D are the Nédélec finite element spaces [30, 31] (also called edge elements) which are known from the context of Maxwell’s equations. Let be a regular triangulation of Ω in tetrahedra in the sense of . The spaces of first kind Nédélec finite elements read
Let . Since , a generalization of (3.4) to 3D seeks with
The discrete exact sequence  implies that the elements in with vanishing Curl are exactly the gradients of functions in . Therefore, the uniqueness in (7.2) can be obtained in the following formulation: Find with
Note that is the kernel of and so (7.3) implies . This variable is introduced in order that (7.3) has the form of a standard mixed system. The discrete Helmholtz decomposition of [1, Lemma 5.4] proves that for the lowest order discretization , is a Crouzeix–Raviart function and so (7.3) can be seen as a generalization of the non-conforming Crouzeix–Raviart FEM to higher polynomial degrees.
The inf-sup condition follows from and . This and the conformity of the method lead to the best-approximation result
Since , this is equivalent to
The following lemma states a projection property similar to Lemma 3.12 for the two-dimensional case. To this end, define
Since is the kernel of , it holds
Let with for all (that means that q is a gradient of an function). Then . If is an admissible refinement of , then .
Since and , the assertion follows with the arguments in the proof of Lemma 3.12. ∎
7.2 Adaptive Algorithm
This subsection outlines the proof of optimal convergence rates for Algorithm 1 in 3D driven by the error estimators λ and μ defined by the local contributions
and (6.2), (6.3). Here, denotes the faces of a tetrahedron and denotes the piecewise constant mesh-size function defined by . The refinement of triangulations in Algorithm 1 is done by newest-vertex bisection . Let denote the space of admissible triangulations with at most N tetrahedra more than . As in Section 6.1, define the seminorm
(Optimal Convergence Rates of AFEM for 3D)
Let . For and sufficiently small and , Algorithm 1 computes sequences of triangulations and discrete solutions for the right-hand side φ of optimal rate of convergence in the sense that
The proof follows as in Section 6 from (A1)–(A4) and (B) from  and the efficiency of λ and μ. The proof of efficiency follows with the standard bubble function technique . The proofs of the axioms (A1)–(A4) and (B) are outlined in the following.
The axioms (A1) stability and (A2) reduction follow as in Section 6.2 with triangle inequalities, inverse inequalities, a trace inequality similar to [10, p. 282], and the mesh-size reduction property for all . However, for (A3) quasi-orthogonality and (A4) discrete reliability, the interpolation operator of  cannot be applied directly to as done in the proof of Theorem 6.8, because . This can be overcome by a quasi-interpolation based on a quasi-interpolation operator from  and a projection operator from . Its properties are summarized in the following theorem.
Let be an admissible refinement of and define
Let . Then there exist , , and with
The differences between the proof of (A4) discrete reliability and the proof of Theorem 6.8 are outlined in the following. Let and denote the discrete solutions to (7.2). As in the proof of Theorem 6.8, let and such that . The first term of the right-hand side of
A piecewise integration by parts and the arguments of the proof of Theorem 6.8 conclude the proof. The crucial point is that is smooth enough to allow for a trace inequality.
The proof of (A3) quasi-orthogonality follows as in the proof of Theorem 6.10 with the projection property of Lemma 7.1 and the following modifications in (6.12). Since (in the analogue notation as in (6.12)) , there exists with . Theorem 7.3 guarantees the existence of , and with . This implies in (6.12) that
Since is smooth enough, a piecewise integration by parts and the arguments of the proof of Theorem 6.8 then prove
This and the arguments of Theorem 6.10 eventually prove the quasi-orthogonality.
8 Numerical Experiments
This section presents numerical experiments for the discretization (3.4) for . Sections 8.1–8.3 compute the discrete solutions on sequences of uniformly red-refined triangulations (see Figure a for a red-refined triangle) as well as on sequences of triangulations created by the adaptive Algorithm 1 with bulk parameter and and . The convergence history plots are logarithmically scaled and display the error against the number of degrees of freedom (ndof) of the linear system resulting from the Schur complement. The underlying L-shaped domain with its initial triangulation is depicted in Figure b.
Red-refined triangle and initial mesh for the L-shaped domain.
8.1 L-Shaped Domain, I
The function u given in polar coordinates by
is harmonic. For the following experiment we choose and with perturbation function ,
such that for . Since , it holds . Let
denote the ball with radius and midpoint . Since and , it holds .
For non-homogeneous Dirichlet data, the jump is defined for boundary edges , , with adjacent triangle by
The errors and error estimators for the approximation of for are plotted in Figure 2 against the number of degrees of freedom. The errors and error estimators show an equivalent behavior with an overestimation of approximately 10. Uniform refinement leads to a suboptimal convergence rate of for . The adaptive refinement reproduces the optimal convergence rates of for . Figure 3 depicts three meshes created by the adaptive algorithm for , 1, and 2 with approximately 1000 degrees of freedom. The singularity at the re-entrant corner leads to a strong refinement towards , while the refinement for also reflects the behavior of the right-hand side, i.e., one also observes a moderate refinement on the circular ring . The marking with respect to the data-approximation ( in Algorithm 1) is applied at the first 7 (resp. 5 and 10) levels for (resp. and ) and then at approximately every third level.
8.2 L-Shaped Domain, II
For and define with .
The error estimators are plotted against the degrees of freedom in Figure 4 for . The error estimators show for a suboptimal convergence rate of for uniform refinement. The adaptive Algorithm 1 recovers the optimal convergence rate of . Adaptively refined meshes are depicted in Figure 5 for approximately 1000 degrees of freedom. The strong refinement towards the singularity at the re-entrant corner is clearly visible. The smoothness of implies that the data-approximation error estimator vanishes on all triangulations for . For , does not vanish, nevertheless, since for all , only the Dörfler marking is applied.
8.3 Singular α
This subsection is devoted to a numerical investigation of the dependence of the error on the regularity of α. The exact smooth solution of
reads . Define with defined by
Then with .
The errors and error estimators are plotted in Figure 6 against the number of degrees of freedom. The convergence rate on uniform red-refined meshes for is and, hence, the convergence rate seems to depend on the regularity of α. The errors and error estimators show the same convergence rate. Figure 7 focuses on the results for and uniform mesh-refinement. The error and the error estimator show a convergence rate between h and , while converges with a rate of due to the singularity of α. This numerical experiment suggests that the error does not depend on the regularity of α (at least in a preasymptotic regime). The triangle inequality implies . This upper bound is also plotted in Figure 7.
Figure 8 depicts adaptively refined meshes for with approximately 1000 degrees of freedom. The singularity of α leads to a strong refinement towards the re-entrant corner. The marking with respect to the data-approximation ( in Algorithm 1) is only applied at levels 1–5, 7, 12, and 18 for . All other marking steps for use the Dörfler marking ().
The author would like to thank Professor C. Carstensen for valuable discussions.
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Published Online: 2016-10-18
Published in Print: 2017-01-01