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Computational Methods in Applied Mathematics

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Volume 17, Issue 1

Issues

A New Generalization of the P 1 Non-Conforming FEM to Higher Polynomial Degrees

Mira Schedensack
Published Online: 2016-10-18 | DOI: https://doi.org/10.1515/cmam-2016-0031

Abstract

This paper generalizes the non-conforming FEM of Crouzeix and Raviart and its fundamental projection property by a novel mixed formulation for the Poisson problem based on the Helmholtz decomposition. The new formulation allows for ansatz spaces of arbitrary polynomial degree and its discretization coincides with the mentioned non-conforming FEM for the lowest polynomial degree. The discretization directly approximates the gradient of the solution instead of the solution itself. Besides the a priori and medius analysis, this paper proves optimal convergence rates for an adaptive algorithm for the new discretization. These are also demonstrated in numerical experiments. Furthermore, this paper focuses on extensions of this new scheme to quadrilateral meshes, mixed FEMs, and three space dimensions.

Keywords: Non-Conforming FEM; Helmholtz Decomposition; Mixed FEM; Adaptive FEM; Optimality; Crouzeix–Raviart FEM; Rannacher–Turek FEM

MSC 2010: 65N30; 65N12; 65N15

1 Introduction

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 P1 non-conforming FEM, also named after Crouzeix and Raviart [21], states that the L2 projection of H01(Ω) onto the space of piecewise constant functions equals the space of piecewise gradients of the non-conforming interpolation of H01(Ω) functions in the P1 non-conforming finite element space. This property is the basis for the proof of the discrete inf-sup condition for the Stokes equations [21] as well as for the analysis of adaptive algorithms [6].

Many possible generalizations of the P1 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 [4] proves equivalence of the novel discretization with the known non-conforming Crouzeix–Raviart FEM [21] 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 [33] 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 u 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 P1 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.

2 Notation

Throughout this paper Ω2 is a simply connected, bounded, polygonal Lipschitz domain. Standard notation on Lebesgue and Sobolev spaces and their norms is employed with L2 scalar product (,)L2(Ω). Given a Hilbert space X, let L2(Ω;X) resp. Hk(Ω;X) denote the space of functions with values in X whose components are in L2(Ω) resp. Hk(Ω), and let L02(Ω) denote the subset of L2(Ω) of functions with vanishing integral mean. The space of L2 functions whose weak divergence exists and is in L2 is denoted with H(div,Ω). The space of infinitely differentiable functions reads C(Ω) and the subspace of functions with compact support in Ω is denoted with Cc(Ω). The piecewise action of differential operators is denoted with a subscript NC. The formula AB represents an inequality ACB for some mesh-size independent, positive generic constant C; AB abbreviates ABA. By convention, all generic constants C1 do neither depend on the mesh-size nor on the level of a triangulation but may depend on the fixed coarse triangulation 𝒯0 and its interior angles. The Curl operator in two dimensions is defined by Curlβ:=(β/x2,-β/x1) for sufficiently smooth β.

A shape-regular triangulation 𝒯 of a bounded, polygonal, open Lipschitz domain Ω2 is a set of closed triangles T𝒯 such that Ω¯=𝒯 and any two distinct triangles are either disjoint or share exactly one common edge or one vertex. Let (T) denote the edges of a triangle T and :=(𝒯):=T𝒯(T) the set of edges in 𝒯. Any edge E is associated with a fixed orientation of the unit normal νE on E (and τE=(0,-1;1,0)νE denotes the unit tangent on E). On the boundary, νE is the outer unit normal of Ω, while for interior edges EΩ, the orientation is fixed through the choice of the triangles T+𝒯 and T-𝒯 with E=T+T- and νE:=νT+|E is then the outer normal of T+ on E. In this situation, [v]E:=v|T+-v|T- denotes the jump across E. For an edge EΩ on the boundary, the jump across E reads [v]E:=v. For T𝒯 and Xn, let Pk(T;X) denote the set of polynomials on a triangle T, and Pk(𝒯;X) the set of piecewise polynomials, i.e.,

Pk(T;X):={v:TXeach component of v is a polynomial of total degreek},Pk(𝒯;X):={v:ΩXT𝒯:v|TPk(T;X)},

and let Pk(𝒯):=Pk(𝒯;). Given a subspace XL2(Ω;n), let ΠX:L2(Ω;n)X denote the L2 projection onto X and let Πk abbreviate ΠPk(𝒯;n). Given a triangle T𝒯, let hT:=(meas2(T))1/2 denote the square root of the area of T and let h𝒯P0(𝒯) denote the piecewise constant mesh-size with h𝒯|T:=hT for all T𝒯. For a set of triangles 𝒯, let abbreviate

:=TL2(T)2.

Given an initial triangulation 𝒯0, an admissible triangulation is a regular triangulation which can be created from 𝒯0 by newest-vertex bisection [40]. The set of admissible triangulations is denoted by 𝕋.

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 P1 non-conforming Crouzeix–Raviart FEM [21].

3.1 New Mixed Formulation of the Poisson Problem

Given the simply connected, bounded, polygonal Lipschitz domain Ω2 and fL2(Ω), the Poisson model problem seeks uH01(Ω) with

-Δu=f in Ωandu=0 on Ω.(3.1)

The novel weak formulation is based on the classical Helmholtz decomposition [35]

L2(Ω;2)=H01(Ω)Curl(H1(Ω)L02(Ω))(3.2)

for any simply connected domain Ω2, where the sum is orthogonal with respect to the L2 scalar product.

Note that for Ω2, the definition of the Curl implies

H(Curl,Ω):={βL2(Ω)CurlβL2(Ω)}=H1(Ω).

Define X:=L2(Ω;2) and Y:=H1(Ω)L02(Ω) and let φH(div,Ω) satisfy -divφ=f. The novel weak formulation of the Poisson problem (3.1) seeks (p,α)X×Y with

(p,q)L2(Ω)+(q,Curlα)L2(Ω)=(φ,q)L2(Ω)for all qX,(3.3a)(p,Curlβ)L2(Ω)=0for all βY.(3.3b)

This formulation is the point of departure for the numerical approximation of u in Section 3.2.

(Existence of Solutions)

Since CurlYX, any βY satisfies the inf-sup condition

CurlβL2(Ω)supqX{0}(q,Curlβ)L2(Ω)qL2(Ω).

This and Brezzi’s splitting lemma [11] imply the unique existence of a solution (p,α)X×Y to (3.3). The L2 orthogonality of p and Curlα implies

pL2(Ω)2+CurlαL2(Ω)2=φL2(Ω)2.

(Equivalence of (3.1) and (3.3))

The second equation of (3.3) and the Helmholtz decomposition (3.2) imply the existence of u~H01(Ω) with p=u~. Since φH(div,Ω) satisfies -divφ=f, the L2 orthogonality in (3.2) implies that any vH01(Ω) satisfies

(p,v)L2(Ω)=(φ,v)L2(Ω)=(f,v)L2(Ω)

and, hence, u~ solves (3.1).

(Mixed Boundary Conditions)

Let Ω=ΓDΓN with ΓD closed, ΓDΓN=, and each connectivity component of ΓD has positive length. Assume that the triangulation resolves ΓD. Let H-1/2(ΓN) denote the space of generalized normal traces of H(div,Ω) functions and let uDH1(Ω) and gH-1/2(ΓN) in the sense that there holds g=qν on ΓN in the sense of distributions for some qH(div,Ω). Consider the mixed boundary value problem -Δu=f in Ω with u|ΓD=uD on ΓD and (uν)|ΓN=g on ΓN. Let HD1(Ω) denote the subspace of H1(Ω) of functions with vanishing trace on ΓD. For ΓD=, define HD1(Ω):=H1(Ω)L02(Ω). Define

H1(Ω):={βYβ is constant on each connectivity component of ΓN}.

The Helmholtz decomposition

L2(Ω;2)=HD1(Ω)CurlH1(Ω)

for mixed boundary conditions [24, Corollary 3.1] then leads to the following formulation. Let φH(div,Ω) with -divφ=f additionally fulfil the boundary condition φν|ΓN=g and seek (p,α)L2(Ω;2)×H1(Ω) with

(p,q)L2(Ω)+(q,Curlα)L2(Ω)=(φ,q)L2(Ω)for all qL2(Ω;2),(p,Curlβ)L2(Ω)=(uD,Curlβ)L2(Ω)for all βH1(Ω).

Since p=φ-CurlαH(div,Ω), the equivalence follows as in Section 3.1 and with

(pν)|ΓN=(φν)|ΓN-(Curlαν)|ΓN=g-(ατ)|ΓN=g.

(Multiply Connected Domains)

If Ω2 is a multiply connected polygonal bounded Lipschitz domain and Ω=ΓDΓN, such that all parts of ΓD lie on the outer boundary of Ω (on the unbounded connectivity component of 2Ω), then the Helmholtz decomposition of Remark 3.4 still holds and a discretization as above is then immediate. However, if the Dirichlet boundary ΓD also covers parts of interior boundary, that Helmholtz decomposition does no longer hold: There exist harmonic functions which are constant on different parts of ΓD and, hence, are neither in HΓD1(Ω), nor in CurlH1(Ω).

(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, φ=(φ1,0) can be defined by a numerical integration of f, e.g., by approximating

φ1(x,y):=Rxf~(t,y)𝑑t

for some R with Ω[-R,)× and f~ defined by f~|Ω=f and f~|([-R,)×)Ω=0. This is possible in parallel.

3.2 Discretization

Let 𝒯 be a regular triangulation of Ω and k{0} and define

Xh(𝒯):=Pk(𝒯;2)andYh(𝒯):=Pk+1(𝒯)Y.

The discretization of (3.3) seeks phXh(𝒯) and αhYh(𝒯) with

(ph,qh)L2(Ω)+(qh,Curlαh)L2(Ω)=(φ,qh)L2(Ω)for all qhXh(𝒯),(3.4a)(ph,Curlβh)L2(Ω)=0for all βhYh(𝒯).(3.4b)

Since there are no continuity conditions on qhXh(𝒯) and since CurlYh(𝒯)Xh(𝒯), equation (3.4a) is fulfilled in a strong form, i.e.,

ph+Curlαh=Πkφ.

In contrast to classical finite element methods, the approximation ph of u is a gradient only in a discrete orthogonal sense, namely (3.4b). For k=0, 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 CurlYh(𝒯)Xh(𝒯), the discrete inf-sup condition

CurlβhL2(Ω)supqhXh(𝒯){0}(qh,Curlβh)L2(Ω)qhL2(Ω)for all βhYh(𝒯)

is fulfilled. This inequality together with Brezzi’s splitting lemma [11] implies the unique existence of a solution (ph,αh)Xh(𝒯)×Yh(𝒯) to (3.4). The equality in

phL2(Ω)2+CurlαhL2(Ω)2=ΠkφL2(Ω)2φL2(Ω)2.

follows from the L2 orthogonality of ph and Curlαh.

The conformity of the method and the inf-sup conditions from Remarks 3.2 and 3.8 imply the following best-approximation result.

(Best-Approximation)

The solution (p,α)X×Y to (3.3) and the discrete solution (ph,αh)Xh(T)×Yh(T) to (3.4) satisfy

p-phL2(Ω)+Curl(α-αh)L2(Ω)minqhXh(𝒯)p-qhL2(Ω)+minβhYh(𝒯)Curl(α-βh)L2(Ω).(3.5)

A direct analysis of the bilinear form :(X×Y)×(X×Y) defined by

((p,α),(q,β)):=(p,q)L2(Ω)+(q,Curlα)L2(Ω)+(p,Curlβ)L2(Ω)

for all p,qX and all α,βY 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

minβhYh(𝒯)Curl(α-βh)L2(Ω)

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 Curlα=φ-u has at least the same regularity as u, 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

minqhXh(𝒯)p-qhL2(Ω)(3.6)

is superior to the best-approximation of a standard FEM, where p=u is approximated with gradients of finite element functions. However, [42, Theorem 3.2] and the comparison results of [15] 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 vH01(Ω), the best-approximation of v in Xh(𝒯) is a discrete orthogonal gradient in the sense that it is orthogonal to CurlYh(𝒯) and so belongs to the set of discrete orthogonal gradients Wh(𝒯) defined by

Wh(𝒯):={qhXh(𝒯)(qh,Curlβh)L2(Ω)=0 for all βhYh(𝒯)}.(3.7)

The projection property is the key ingredient in the optimality analysis of Section 6.

(Projection Property)

It holds that Wh(T)=ΠXh(T)H01(Ω). Moreover, if T is an admissible refinement of T, then ΠXh(T)Wh(T)=Wh(T).

Proof.

Let qH01(Ω). Since CurlYh(𝒯)Xh(𝒯) and Yh(𝒯)Y, the orthogonality in the Helmholtz decomposition (3.2) implies for any βhYh(𝒯) that

(ΠXh(𝒯)q,Curlβh)L2(Ω)=(q,Curlβh)L2(Ω)=0.

This proves ΠXh(𝒯)H01(Ω)Wh(𝒯). For the converse direction, let phWh(𝒯) and let uH01(Ω) be a solution (possibly not unique) to

(ΠXh(𝒯)u,ΠXh(𝒯)v)L2(Ω)=(ph,ΠXh(𝒯)v)L2(Ω)for all vH01(Ω).

The orthogonality of ph-ΠXh(𝒯)u to H01(Ω) implies the existence of αY such that ph-ΠXh(𝒯)u=Curlα. Therefore, CurlαXh(𝒯) and, hence, α is a piecewise polynomial of degree k+1 and therefore αYh(𝒯). But since phWh(𝒯), it holds that

CurlαL2(Ω)2=(ph-ΠXh(𝒯)u,Curlα)L2(Ω)=0

and, hence, α=0. This proves ΠXh(𝒯)u=ph and, therefore, Wh(𝒯)ΠXh(𝒯)H01(Ω).

A similar proof applies in the discrete case and proves ΠXh(𝒯)Wh(𝒯)=Wh(𝒯). ∎

(Computational Costs)

Problem (3.4) is equivalent to: Find (ph,αh)Xh(𝒯)×Yh(𝒯) such that

(Curlβh,Curlαh)L2(Ω)=(φ,Curlβh)L2(Ω)for all βhYh(𝒯),ph=ΠXh(𝒯)φ-Curlαh.

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 [21] reads

CR01(𝒯):={vCRP1(𝒯)vCR is continuous at midpoints of interior edges  and vanishes at midpoints of boundary edges}.

Since CR01(𝒯)H01(Ω) (if the triangulation consists of more than one triangle), the weak gradient of a function vCRCR01(𝒯) does not exist in general. However, the piecewise version NCvCRP0(𝒯;2) defined by (NCvCR)|T:=(vCR|T) for all T𝒯 exists. The P1 non-conforming discretization of the Poisson problem seeks uCRCR01(𝒯) with

(NCuCR,NCvCR)L2(Ω)=(f,vCR)L2(Ω)for all vCRCR01(𝒯).(3.8)

The lowest-order space of Raviart–Thomas finite element functions [34] reads

RT0(𝒯):={qRTH(div,Ω)T𝒯aT2,bT with qRT(x)=aT+bTx}.

The Raviart–Thomas functions have the property that the integration by parts formula holds for functions in H01(Ω) as well as for functions in CR01(𝒯).

The following proposition proves the equivalence of the P1 non-conforming discretization and the discretization (3.4) for k=0. Note that the discretization (3.8) is a non-conforming discretization, while the discretization (3.4) is a conforming one.

(Equivalence with CR-NCFEM)

Let fP0(T) be piecewise constant and let φRTRT0(T) satisfy -divφRT=f. Then the discrete solution (ph,αh)P0(T;R2)×(P1(T)Y) to (3.4) for k=0 and the gradient of the discrete solution uCRCR01(T) to (3.8) coincide,

ph=NCuCR.(3.9)

Proof.

The crucial point is the discrete Helmholtz decomposition [4]

P0(𝒯;2)=NCCR01(𝒯)Curl(P1(𝒯)Y).(3.10)

Since ph is L2 orthogonal to Curl(P1(𝒯)Y), this implies ph=NCu~CR for some u~CRCR01(𝒯). Let qh=NCvCR for some vCRCR01(𝒯). Then qh is L2 orthogonal to Curl(P1(𝒯)Y) and a piecewise integration by parts and (3.4) imply

(NCu~CR,NCvCR)L2(Ω)=(ph,qh)L2(Ω)=(φRT,qh)L2(Ω)=(-divφRT,vCR)L2(Ω)=(f,vCR)L2(Ω).

Hence, u~CR=uCR solves (3.8). ∎

The projection property from Lemma 3.12 generalizes the famous integral mean property

NCINCv=ΠP0(𝒯;2)vfor all vH01(Ω)

of the non-conforming interpolation operator INC.

(Higher Polynomial Degrees)

For higher polynomial degrees k1, the discretization (3.4) is not equivalent to known non-conforming schemes [23, 20, 21, 28], in the sense that Wh(𝒯)NCVh(𝒯) for those non-conforming finite element spaces Vh(𝒯). This follows from NCVh(𝒯)Wh(𝒯) for non-conforming FEMs with enrichment. A dimension argument shows

dim(Wh(𝒯))>dimVh(𝒯)

for the non-conforming FEMs of [23, 20] without enrichment and therefore Wh(𝒯)NCVh(𝒯). 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).

4 Extensions

Section 4.1 generalizes the novel FEM to quadrilateral meshes and proves a new discrete Helmholtz decomposition for the Q1 rotated non-conforming Rannacher–Turek FEM [33]. Section 4.2 discusses a discretization with Raviart–Thomas functions.

4.1 Quadrilateral Finite Elements

For this subsection, consider a regular partition 𝒯 of Ω in quadrilaterals. Define, for the reference rectangle T^=[0,1]2,

Qk(T^):={vhP2k(T^)f,gPk([0,1]):vh(x,y)=f(x)g(y)}.

Given T𝒯, let ψT:T^T denote the bilinear transformation from the reference rectangle to T. For consistency, let P-1([0,1]):={0} and set

VQ,k(𝒯):={βhYT𝒯:(βhψT)|T^Qk(T^)},

Xkrect(T^):={τhL2(T^;2)a,b,cPk-2([0,1]),d,eQk-1(T^) such that (x^,y^)T^:       τh(x^,y^)=a(-x^ky^k-1x^k-1y^k)+(x^kb(y^)+d(x^,y^)y^kc(x^)+e(x^,y^))},Xkrect(𝒯):={τhL2(Ω;2)T𝒯:ρTXkrect(T^) such that       (τhψT)|T^=(01-10)D(ψT-1)ψT(0-110)ρT}.

Then a discretization with respect to the quadrilateral partition seeks phXkrect(𝒯) and αhVQ,k(𝒯) with

(ph,qh)L2(Ω)+(qh,Curlαh)L2(Ω)=(φ,qh)L2(Ω) for all qhXkrect(𝒯),(ph,Curlβh)L2(Ω)=0 for all βhVQ,k(𝒯).

Let βhVQ,k(𝒯), i.e., (βhψT)|T^Qk(T^). A direct calculation reveals, for all T𝒯,

((Curlβh)ψT)|T=((βhψTψT-1))ψT(01-10)=((βhψT))D(ψT-1)ψT(01-10).

Let (βhψT)(x^,y^)=(𝔞x^k+f(x^))(𝔟y^k+g(y^)) with 𝔞,𝔟 and f,gPk-1([0,1]). Then it holds

(βhψT)=𝔞𝔟k(x^k-1y^ky^k-1x^k)+(𝔟y^kf(x^)/x^𝔞x^kg(y^)/y^)+(𝔞kx^k-1g(y^)+g(y^)f(x^)/x^𝔟ky^k-1f(x^)+f(x^)g(y^)/y^)

and therefore

((βhψT))(0-110)=(𝔞𝔟k(y^k-1x^k-x^k-1y^k)+(𝔞x^kg(y^)/y^-𝔟y^kf(x^)/x^)+(𝔟ky^k-1f(x^)+f(x^)g(y^)/y^-𝔞kx^k-1g(y^)-g(y^)f(x^)/x^))=:(ρT(x^,y^)).

This implies ρTXkrect(T^) and ((βhψT))=(0,1;-1,0)ρT. The combination of the previous equalities leads to

((Curlβh)ψT)|T=(01-10)D(ψT-1)ψT(0-110)ρT.

Consequently, CurlβhXkrect(𝒯). This and the conformity of the method prove as in Section 3 the following statements:

  • (i)

    unique existence of solutions,

  • (ii)

    the best-approximation result

    p-phL2(Ω)+Curl(α-αh)L2(Ω)(minqhXkrect(𝒯)p-qhL2(Ω)+minβhVQ,k(𝒯)Curl(α-βh)L2(Ω)),

  • (iii)

    the projection property

    ΠXkrect(𝒯)H01(Ω)Whrect(𝒯)

    for

    Whrect(𝒯)={qhXkrect(𝒯)βhVQ,k(𝒯):(qh,Curlβh)L2(Ω)=0}.(4.1)

The properties (i)–(iii) still hold for any X~h(𝒯) with Xkrect(𝒯)X~h(𝒯)X.

The remaining part of this subsection proves the equivalence of the lowest-order rectangular discretization with the non-conforming Rannacher–Turek FEM [33]. To this end, define for the reference rectangle T^ and the bilinear transformation ψT:T^T,

Qrot(T^):=span{1,x,y,x2-y2},VNCrot(𝒯):={vhL2(Ω)T𝒯:(vhψT)|T^Qrot(T^) and Evhds is continuous     for all interior edges E and vanishes at boundary edges E}.(4.2)

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

card()+card((Ω))=3card(𝒯),card((Ω))+card(𝒩)=2card(𝒯)+1(4.3)

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 T be a regular partition of Ω in quadrilaterals with edges E, interior edges E(Ω), and vertices N. Then it holds that 3card(T)+1=card(E(Ω))+card(N).

Proof.

Define a triangulation 𝒯Δ of Ω in triangles by the division of each quadrilateral into two triangles by a diagonal cut. Let Δ denote the edges of 𝒯Δ, Δ(Ω) the interior edges and 𝒩Δ the vertices. Then the following relations between the two partitions hold:

card(𝒯Δ)=2card(𝒯),card(Δ)=card()+card(𝒯),card(Δ(Ω))=card((Ω))+card(𝒯),card(𝒩Δ)=card(𝒩).

This and Euler’s formulae for triangles (4.3) prove

card((Ω))+card(𝒩)=card(Δ(Ω))-card(𝒯)+card(𝒩Δ)=2card(𝒯Δ)+1-card(𝒯)=3card(𝒯)+1.

The following theorem proves that the solution space Whrect(𝒯) from (4.1) equals the piecewise gradients of functions in VNCrot(𝒯) on a partition in squares for k=1.

(Discrete Helmholtz Decomposition on Squares)

Let T be a regular partition of Ω in squares. Then,

X1rect(𝒯)=NCVNCrot(𝒯)CurlVQ,1(𝒯)(4.4)

and the decomposition is L2 orthogonal.

The L2 orthogonality in (4.4) still holds for a partition in parallelograms. However,

NCVNCrot(𝒯)X1rect(𝒯)

for general quadrilateral partitions.

Proof of Theorem 4.3.

Let vhVNCrot(𝒯) and βhVQ,1(𝒯). A piecewise integration by parts leads to

(NCvh,Curlβh)L2(Ω)=EE[vh]EβhτEds.

Since 𝒯 consists of parallelograms, the bilinear transformation ψT:T^T is affine and, hence, βh|E is affine on each edge E. This implies that βhτE is constant. Since the integral mean of [vh]E vanishes, this proves the L2 orthogonality.

Let vhVNCrot(𝒯). A computation reveals for all T𝒯 that there exist fT and gT2 such that

vh(x,y)=D(ψT-1)(fT(-xy)+gT).

For k=1, X1rect(𝒯) reads

X1rect(𝒯)={τhL2(Ω;2)T𝒯:aT,dT2 such that  (τhψT)|T^=(01-10)(D(ψT-1)ψT)(0-110)(aT(-xy)+dT)}.

Since all T𝒯 are squares, DψT and (0,1;-1,0) commute, and, hence, vhX1rect(𝒯). Thus,

NCVNCrot(𝒯)CurlVQ,1(𝒯)X1rect(𝒯).

The dimension of NCVNCrot(𝒯) equals card((Ω)) and the dimension of CurlVQ,1(𝒯) equals card(𝒩)-1, while the dimension of X1rect(𝒯) equals 3card(𝒯). This and Lemma 4.2 prove the assertion. ∎

(Arbitrary Quadrilaterals)

The best-approximation (ii) from above proves quasi-optimal convergence even for arbitrary quadrilaterals. Standard interpolation error estimates for VQ,1(𝒯) and for P0(𝒯;2)X1rect(𝒯) (see [19]) lead to first-order convergence rates of h for sufficiently smooth solutions. This should be contrasted with [33], where quasi-optimal convergence is only obtained for a modification of (4.2) where VNCrot(𝒯) 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 [34] can be regarded as a particular choice of the ansatz spaces in the new mixed scheme.

Let 𝒯 denote a regular triangulation of Ω in triangles. Define the space of Raviart–Thomas functions [34]

XRT(𝒯)={qRTH(div,Ω)T𝒯:qRT|T(x)Pk(T;2)+Pk(T)x}

and

YRT(𝒯):=Pk+1(𝒯)Y.

Then the following problem is a discretization of (3.3): Find (pRT,αRT)XRT(𝒯)×YRT(𝒯) with

(pRT,qRT)L2(Ω)+(qRT,CurlαRT)L2(Ω)=(φ,qRT)for all qRTXRT(𝒯),(4.5a)(pRT,CurlβRT)L2(Ω)=0for all βRTYRT(𝒯).(4.5b)

Since CurlYRT(𝒯)Pk(𝒯;2) and divCurlvRT=0 for all vRTYRT(𝒯), it follows CurlYRT(𝒯)XRT(𝒯). This and the conformity of the method guarantee as in Sections 3 and 4.1 the unique existence of solutions, a best-approximation result, and the projection property

ΠXRT(𝒯)H01(Ω)WRT(𝒯):={qRTXRT(𝒯)βRTYRT(𝒯):(qRT,CurlβRT)L2(Ω)=0}.

The discrete Helmholtz decomposition of [26, 5, 12] proves

XRT(𝒯)=RTPk(𝒯)CurlYRT(𝒯)

with the operator RT:Pk(𝒯)XRT(𝒯) defined for all vRTPk(𝒯) by

(RTvRT,qRT)L2(Ω)=-(vRT,divqRT)L2(Ω)for all qRTXRT(𝒯).

This decomposition yields the equivalence of (4.5) with the problem: Find (pRT,u~RT)XRT(𝒯)×Pk(𝒯) with

pRT=RTu~RT,(wRT,divpRT)L2(Ω)=(divΠXRT(𝒯)φ,wRT)L2(Ω)for all wRTPk(𝒯).

This is the classical Raviart–Thomas discretization with f replaced by divΠXRT(𝒯)φ.

Assume now that the right-hand side φXRT(𝒯) is a Raviart–Thomas function. Since by definition YRT(𝒯)=Yh(𝒯) with Yh(𝒯) from Section 3.2 and since αRT is the solution of

(CurlβRT,CurlαRT)L2(Ω)=(φ,CurlβRT)L2(Ω)for all βRTYRT(𝒯),

it holds αRT=αh with αh from (3.4). Since φ=pRT+CurlαRT and ΠXh(𝒯)φ=ph+Curlαh, it follows

ph=ΠXh(𝒯)pRT.

For k=0, the equivalence with the Crouzeix–Raviart FEM (3.9) then proves the identity

NCuCR=ΠXh(𝒯)pRT,

which is also known as Marini identity [3, 27].

5 Medius Analysis

The medius analysis of [25, 15] proves for the discrete solution uCRCR01(𝒯) to (3.8) the best-approximation result

NC(u-uCR)L2(Ω)minvCRCR01(𝒯)NC(u-vCR)L2(Ω)+osc(f,𝒯).(5.1)

The following theorem proves a generalization for the discretization (3.4) for the lowest order case k=0.

(Best-Approximation Property)

Let (p,α)X×Y be the solution to (3.3) and let (ph,αh)P0(T;R2)×(P1(T)Y) be the solution to (3.4). Then the following best-approximation result holds:

p-phL2(Ω)p-Π0pL2(Ω)+osc(f,𝒯)+supvCRCR01(𝒯){0}(f,vCR)L2(Ω)-(φ,NCvCR)L2(Ω)NCvCRL2(Ω).(5.2)

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).

The remaining part of this section is devoted to the proof of Theorem 5.1. The following lemma from [17, 14] is the key ingredient of this proof. Recall the definition of CR01(𝒯) from Section 3.3.

(Companion)

For any vCRCR01(T) there exists vH01(Ω) with the following properties:

  • (i)

    Π0NC(v-vCR)=0.

  • (ii)

    Π0(v-vCR)=0.

  • (iii)

    h𝒯-1(vCR-v)L2(Ω)+NC(vCR-v)L2(Ω)NCvCRL2(Ω).

Proof of Theorem 5.1.

Define qh:=Π0p-phP0(𝒯;2). The projection property of Lemma 3.12 implies that qhWh(𝒯) and the discrete Helmholtz decomposition (3.10) guarantees the existence of vCRCR01(𝒯) with qh=NCvCR. Let vH01(Ω) denote the companion of vCR from Lemma 5.3. Then

(p-ph,qh)L2(Ω)=(p,NC(vCR-v))L2(Ω)+(p,v)L2(Ω)-(ph,NCvCR)L2(Ω).(5.3)

For the first term on the right-hand side, properties (i) and (iii) from Lemma 5.3 yield

(p,NC(vCR-v))L2(Ω)=(p-Π0p,NC(vCR-v))L2(Ω)p-Π0pL2(Ω)NCvCRL2(Ω).(5.4)

For the second and third term on the right-hand side of (5.3), problems (3.3) and (3.4) lead to

(p,v)L2(Ω)-(ph,NCvCR)L2(Ω)=(φ,v)L2(Ω)-(φ,NCvCR)L2(Ω).

Since -divφ=f, it follows

(φ,v)L2(Ω)-(φ,NCvCR)L2(Ω)=(f,v-vCR)L2(Ω)+(f,vCR)L2(Ω)-(φ,NCvCR)L2(Ω).

Properties (ii) and (iii) of Lemma 5.3 prove

(f,v-vCR)L2(Ω)osc(f,𝒯)NCvCRL2(Ω).

The combination with (5.3) and (5.4) and a Cauchy inequality yield

(p-ph,qh)L2(Ω)(p-Π0pL2(Ω)+osc(f,𝒯)+supvCRCR01(𝒯){0}(f,vCR)L2(Ω)-(φ,NCvCR)L2(Ω)NCvCRL2(Ω))qhL2(Ω).

This and

p-phL2(Ω)2=p-Π0pL2(Ω)2+qhL2(Ω)2=p-Π0pL2(Ω)2+(p-ph,qh)L2(Ω)

prove the assertion. ∎

(Higher Polynomial Degrees)

For k1, Remark 3.15 implies that an analogue of Lemma 5.3 cannot be proved in the same way.

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 𝒯0 denote some initial shape-regular triangulation of Ω, such that each triangle T𝒯 is equipped with a refinement edge ET(T). A proper choice of these refinement edges guarantees an overhead control [7].

Let 𝕋(N) denote the subset of 𝕋 of all admissible triangulations with at most card(𝒯0)+N triangles. The adaptive algorithm involves the overlay of two admissible triangulations 𝒯,𝒯𝕋, which reads

𝒯𝒯:={T𝒯𝒯K𝒯,K𝒯 with TKK}.

Given a triangulation 𝒯, define for all T𝒯 the local error estimator contributions by

λ2(𝒯,T):=h𝒯curlNCphL2(T)2+hTE(T)[ph]EτEL2(E)2andμ2(T):=φ-ΠkφL2(T)2(6.1)

and the global error estimators by

λ2:=λ2(𝒯,𝒯)andμ2:=μ2(𝒯)(6.2)

with

λ2(𝒯,):=Tλ2(𝒯,T)andμ2():=Tμ2(T)(6.3)

for any 𝒯. The adaptive algorithm is driven by these two error estimators and runs the following loop.

(AFEM.)

(Separate Versus Collective Marking)

The residual-based error estimator λ2+μ2 involves the term φ-ΠkφL2(T) without a multiplicative positive power of the mesh-size. Therefore, the optimality of an adaptive algorithm based on collective marking (that is κ= and λ replaced by λ2+μ2 in Algorithm 1) does not follow from the abstract framework from [13]. The reduction property (axiom (A2) from [13]), 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 (μ2>κλ2) can be realized by the algorithm Approx from [7, 16], i.e., the thresholding second algorithm [8] 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. [16]. For a discussion about other algorithms that realize Mark in the second case, see again [16].

For s>0 and (p,α,φ)X×Y×H(div,Ω) define

|(p,α,φ)|𝒜s:=supN0Nsinf𝒯𝕋(N)(p-ΠXh(𝒯)pL2(Ω)+infβ𝒯Yh(𝒯)Curl(α-β𝒯)L2(Ω)+φ-ΠXh(𝒯)φL2(Ω)).

(Pure Local Approximation Class)

Since Ω is assumed to be a Lipschitz domain, all patches in an admissible triangulation 𝒯𝕋 are edge-connected, i.e., for all vertices z𝒩 and triangles T,K𝒯 with zTK, there exists m0 and K0,,Km𝒯 with K0=T, Km=K, zK0Km and Kj-1Kj for all 1jm. Under this assumption, [42, Theorem 3.2] shows

minvhPk+1(𝒯)H1(Ω)(v-vh)L2(Ω)v-ΠkvL2(Ω)for all vH1(Ω).

Hence,

|(p,α,φ)|𝒜s|(p,α,φ)|𝒜s:=supN0Nsinf𝒯𝕋(N)(p-ΠXh(𝒯)pL2(Ω)+Curlα-ΠXh(𝒯)CurlαL2(Ω)+φ-ΠXh(𝒯)φL2(Ω)).

In the following, we assume that the following assumption (B1) holds for the algorithm used in the step Mark for μ2>κλ2 (see Remark 6.2).

((B1) Optimal Data Approximation)

Assume that |(p,α,φ)|𝒜σ is finite. Given a tolerance Tol, the algorithm used in Mark in the second case (μ2>κλ2) in Algorithm 1 computes 𝒯𝕋 with

card(𝒯)-card(𝒯0)Tol-1/(2σ)andμ2(𝒯)Tol.

The following theorem states optimal convergence rates of Algorithm 1.

(Optimal Convergence Rates of AFEM)

For 0<ρB<1 and sufficiently small 0<κ and 0<θ<1, Algorithm 1 computes sequences of triangulations (T)N and discrete solutions (p,α)N for the right-hand side φ of optimal rate of convergence in the sense that

(card(𝒯)-card(𝒯0))s(p-pL2(Ω)+Curl(α-α)L2(Ω))|(p,α,φ)|𝒜s.

The proof follows from the abstract framework of [16], which employs the bounded overhead [7] of the newest-vertex bisection, under the assumptions (A1)–(A4) and (B2) and (SA) which are proved in Sections 6.26.5.

6.2 (A1) Stability and (A2) Reduction

The following two theorems follow from the structure of λ.

(Stability)

Let T be an admissible refinement of T and MTT. Let (pT,αT)Xh(T)×Yh(T) and (pT,αT)Xh(T)×Yh(T) be the respective discrete solutions to (3.4). Then,

|λ(𝒯,)-λ(𝒯,)|p𝒯-p𝒯L2(Ω).

Proof.

This follows with triangle inequalities, inverse inequalities and the trace inequality from [10, p. 282] as in [18, Proposition 3.3]. ∎

(Reduction)

Let T be an admissible refinement of T. Then there exist 0<ρ2<1 and Λ2< such that

λ2(𝒯,𝒯𝒯)ρ2λ2(𝒯,𝒯𝒯)+Λ2p𝒯-p𝒯L2(Ω)2.

Proof.

This follows with a triangle inequality and the mesh-size reduction property h𝒯2|Th𝒯2|T/2 for all T𝒯𝒯 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.

(Discrete Reliability)

Let T be an admissible refinement of T with respective discrete solutions (pT,αT)Xh(T)×Yh(T) and (pT,αT)Xh(T)×Yh(T). Then,

p𝒯-p𝒯L2(Ω)2+Curl(α𝒯-α𝒯)L2(Ω)2λ2(𝒯,𝒯𝒯)+μ2(𝒯,𝒯𝒯).

Proof.

Recall the definition of Wh(𝒯) from (3.7). Since p𝒯-p𝒯Xh(𝒯), there exist σ𝒯Wh(𝒯) and r𝒯Yh(𝒯) with p𝒯-p𝒯=σ𝒯+Curlr𝒯. Since Wh(𝒯)L2(Ω)CurlYh(𝒯), it holds

σ𝒯L2(Ω)2+Curlr𝒯L2(Ω)2=p𝒯-p𝒯L2(Ω)2.

The orthogonality furthermore implies that the discrete error can be split as

p𝒯-p𝒯L2(Ω)2=(p𝒯-p𝒯,σ𝒯)L2(Ω)+(p𝒯-p𝒯,Curlr𝒯)L2(Ω).

The projection property, Lemma 3.12, proves ΠXh(𝒯)σ𝒯Wh(𝒯). Hence, problem (3.4) implies that the first term of the right-hand side equals

(p𝒯-p𝒯,σ𝒯)L2(Ω)=(ΠXh(𝒯)φ-φ,σ𝒯)L2(Ω)=(ΠXh(𝒯)φ-ΠXh(𝒯)φ,σ𝒯)L2(Ω).

For any triangle T𝒯𝒯, it holds (ΠXh(𝒯)φ-ΠXh(𝒯)φ)|T=0. Therefore,

(ΠXh(𝒯)φ-ΠXh(𝒯)φ,σ𝒯)L2(Ω)ΠXh(𝒯)φ-ΠXh(𝒯)φ𝒯𝒯σ𝒯L2(Ω).

Since 𝒯 is a refinement of 𝒯, it holds

ΠXh(𝒯)φ-ΠXh(𝒯)φ𝒯𝒯=ΠXh(𝒯)(ΠXh(𝒯)φ-φ)𝒯𝒯φ-ΠXh(𝒯)φ𝒯𝒯.

Let r𝒯Yh(𝒯) denote the quasi interpolant from [39] of r𝒯 which satisfies the approximation and stability properties

h𝒯-1(r𝒯-r𝒯)L2(Ω)+Curl(r𝒯-r𝒯)L2(Ω)Curlr𝒯L2(Ω)

and (r𝒯)|E=(r𝒯)|E for all edges E(𝒯)(𝒯). Since p𝒯Wh(𝒯) and p𝒯Wh(𝒯), it holds

(p𝒯-p𝒯,Curlr𝒯)L2(Ω)=(p𝒯,Curl(r𝒯-r𝒯))L2(Ω).

An integration by parts leads to

(p𝒯,Curl(r𝒯-r𝒯))L2(Ω)=-(curlNCp𝒯,r𝒯-r𝒯)L2(Ω)+E(𝒯)E[p𝒯τE]E(r𝒯-r𝒯)𝑑s.(6.4)

For a triangle T𝒯𝒯, any edge E(T) satisfies E(𝒯)(𝒯). Hence, (r𝒯)|T=(r𝒯)|T for all triangles T𝒯𝒯. This, the Cauchy inequality and the approximation and stability properties of the quasi interpolant lead to

-(curlNCp𝒯,r𝒯-r𝒯)L2(Ω)h𝒯curlNCp𝒯𝒯𝒯Curlr𝒯L2(Ω).(6.5)

Since (r𝒯)|E=(r𝒯)|E for all edges E(𝒯)(𝒯), the approximation and stability properties of the quasi interpolant and the trace inequality [10, p. 282] lead to

EE[p𝒯τE]E(r𝒯-r𝒯)𝑑sE(𝒯)(𝒯)hT[p𝒯τE]EL2(E)2Curlr𝒯L2(Ω).(6.6)

The combination of the previous displayed inequalities yields

p𝒯-p𝒯L2(Ω)2λ2(𝒯,𝒯𝒯)+μ2(𝒯,𝒯𝒯).

Since Curlα𝒯=ΠXh(𝒯)φ-p𝒯 and Curlα𝒯=ΠXh(𝒯)φ-p𝒯, 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)

Let (p,α)X×Y and (ph,αh)Xh(T)×Yh(T) be the solutions to (3.3) and (3.4) for some TT. There exist constants Ceff,Crel>0 with

Ceff-2(λ2(𝒯,𝒯)+μ2(𝒯))p-phL2(Ω)2+Curl(α-αh)L2(Ω)2Crel2(λ2(𝒯,𝒯)+μ2(𝒯)).

Proof.

The a priori error estimate from Theorem 3.9 implies the convergence of the discrete solutions. This and Theorem 6.8 prove the reliability. The efficiency follows from the standard bubble function technique [43]. ∎

6.4 (A3) Quasi-Orthogonality

The following theorem proves quasi-orthogonality of the discretization (3.4).

(General Quasi-Orthogonality)

Let (TjjN) be some sequence of triangulations with discrete solutions (pj,αj)Xh(Tj)×Yh(Tj) to (3.4). Let N. Then,

j=(pj-pj-1L2(Ω)2+Curl(αj-αj-1)L2(Ω)2)λ-12+μ-12.

Proof.

The projection property, Lemma 3.12, proves ΠXh(𝒯j-1)pjWh(𝒯j-1) with Wh(𝒯j-1) from (3.7). Hence, problem (3.4) leads to

(pj-1,pj-pj-1)L2(Ω)=(φ,ΠXh(𝒯j-1)pj-pj-1)L2(Ω),(pj,pj-pj-1)L2(Ω)=(φ,pj)-(φ,ΠXh(𝒯j-1)pj)L2(Ω).

The subtraction of these two equations and an index shift lead, for any M with M>, to

j=Mpj-pj-1L2(Ω)2=j=M(φ,pj-ΠXh(𝒯j-1)pj)L2(Ω)-j=M(φ,ΠXh(𝒯j-1)pj)L2(Ω)+j=-1M-1(φ,pj)L2(Ω)=(φ,p-1-pM)L2(Ω)+2j=M(φ,pj-ΠXh(𝒯j-1)pj)L2(Ω).(6.7)

Since pj-ΠXh(𝒯j-1)pjXh(𝒯j) is L2 orthogonal to Xh(𝒯j-1), a Cauchy and a weighted Young inequality imply

2j=M(φ,pj-ΠXh(𝒯j-1)pj)L2(Ω)=2j=M(ΠXh(𝒯j)φ-ΠXh(𝒯j-1)φ,pj-ΠXh(𝒯j-1)pj)L2(Ω)2j=MΠXh(𝒯j)φ-ΠXh(𝒯j-1)φL2(Ω)2+12j=Mpj-ΠXh(𝒯j-1)pjL2(Ω)2.(6.8)

The orthogonality ΠXh(𝒯j)φ-ΠXh(𝒯j-m)φL2(Ω)Xh(𝒯j-m) for all 0mj proves

j=MΠXh(𝒯j)φ-ΠXh(𝒯j-1)φL2(Ω)2=ΠXh(𝒯M)φ-ΠXh(𝒯-1)φL2(Ω)2.(6.9)

The definition of μ yields

ΠXh(𝒯M)φ-ΠXh(𝒯-1)φL2(Ω)=ΠXh(𝒯M)(φ-ΠXh(𝒯-1)φ)L2(Ω)μ-1.(6.10)

The combination of (6.7)–(6.10) and pj-ΠXh(𝒯j-1)pjL2(Ω)pj-pj-1L2(Ω) leads to

12j=Mpj-pj-1L2(Ω)22μ-12+(φ,p-1-pM)L2(Ω).(6.11)

The combination of the arguments of (6.4), (6.5) and (6.6) proves

(Curl(αM-α-1),p-1)L2(Ω)λ-1Curl(αM-α-1)L2(Ω).(6.12)

This, the discrete problem (3.4), and the discrete reliability Curl(αM-α-1)L2(Ω)λ-1+μ-1 from Theorem 6.8 lead to

(p-1-pM,ΠXh(𝒯-1)φ)L2(Ω)=(p-1-pM,p-1+Curlα-1)L2(Ω)=(p-1-pM,p-1)L2(Ω)=(Curl(αM-α-1),p-1)L2(Ω)λ-1Curl(αM-α-1)L2(Ω)(λ-1+μ-1)2.

This and a further application of Theorem 6.8 lead to

(φ,p-1-pM)L2(Ω)=(φ-ΠXh(𝒯-1)φ,p-1-pM)L2(Ω)+(p-1-pM,ΠXh(𝒯-1)φ)L2(Ω)φ-ΠXh(𝒯-1)φL2(Ω)p-1-pML2(Ω)+(λ-1+μ-1)L2(Ω)2(λ-1+μ-1)2.(6.13)

The combination of (6.11) with (6.13) implies

j=Mpj-pj-1L2(Ω)2λ-12+μ-12.(6.14)

The Young inequality, the triangle inequality, and Curlαj=ΠXh(𝒯j)φ-pj imply

j=MCurl(αj-αj-1)L2(Ω)22j=Mpj-pj-1L2(Ω)2+2j=MΠXh(𝒯j)φ-ΠXh(𝒯j-1)φL2(Ω)2.

Since M> is arbitrary, the combination with (6.9), (6.10), and (6.14) yields the assertion. ∎

6.5 (B) Data Approximation

The following theorem together with Assumption 6.4 forms the axiom (B) from [16].

((B2) Quasimonotonicity and (SA) Sub-Additivity)

Any admissible refinement T of T satisfies

μ2(𝒯)μ2(𝒯)𝑎𝑛𝑑T𝒯,TKμ2(T)μ2(K)for all K𝒯.

Proof.

This follows directly from the definition of μ. ∎

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 Ω3 be a simply connected, bounded, polygonal Lipschitz domain in 3. For the sake of simplicity, we also assume that Ω is connected (i.e., Ω is contractible). The Curl operator acts on a sufficiently smooth vector field β:Ω3 as Curlβ=β with the cross product or vector product . Let H(Curl,Ω) denote the space of all βL2(Ω;3) with CurlβL2(Ω;3) for the weak Curl, i.e.,

ΩvCurlβdx=ΩβCurlvdxfor all vCc(Ω;3).

In contrast to the two-dimensional case, H(Curl,Ω)H1(Ω;3). The Helmholtz decomposition in 3D reads

L2(Ω;3)=H01(Ω)CurlH(Curl,Ω)

and the sum is L2 orthogonal. It is a consequence of the identity

{rH(div,Ω)divr=0}=CurlH(Curl,Ω)

in the De Rham complex [9].

Let φH(div,Ω) with -divφ=f. Then the Poisson problem (3.1) is equivalent to the problem: Find (p,α)L2(Ω;3)×H(Curl,Ω) with

(p,q)L2(Ω)+(q,Curlα)L2(Ω)=(φ,q)L2(Ω)for all qL2(Ω;3),(7.1a)(p,Curlβ)L2(Ω)=0for all βH(Curl,Ω).(7.1b)

In contrast to the two-dimensional case, the operator Curl:H(Curl,Ω)L2(Ω;3) has a non-trivial kernel. Classical results [35] characterize this kernel as H1(Ω). To enforce uniqueness, we can reformulate (7.1) as follows: Find (p,α,w)L2(Ω;3)×H(Curl,Ω)×(H1(Ω)L02(Ω)) with

(p,q)L2(Ω)+(q,Curlα)L2(Ω)=(φ,q)L2(Ω)for all qL2(Ω;3),(p,Curlβ)L2(Ω)+(β,w)L2(Ω)=0for all βH(Curl,Ω),(α,v)L2(Ω)=0for all v(H1(Ω)L02(Ω)).

Note that {βH(Curl,Ω)Curlβ=0}=H1(Ω) implies w=0.

Standard finite element spaces to discretize H(Curl,Ω) 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 𝒯 be a regular triangulation of Ω in tetrahedra in the sense of [19]. The spaces of first kind Nédélec finite elements read

YN,k(T):=Pk(T;3)+(xPk(T;3)),YN,k(𝒯):={βhH(Curl,Ω)T𝒯:βh|TYN,k(T)}.

Let Xh(𝒯):=Pk(𝒯;3). Since CurlYN,k(𝒯)Xh(𝒯), a generalization of (3.4) to 3D seeks (ph,αh)Xh(𝒯)×YN,k(𝒯) with

(ph,qh)L2(Ω)+(qh,Curlαh)L2(Ω)=(φ,qh)L2(Ω)for all qhXh(𝒯),(7.2a)(ph,Curlβh)L2(Ω)=0for all βhYN,k(𝒯).(7.2b)

The discrete exact sequence [9] implies that the elements in YN,k(𝒯) with vanishing Curl are exactly the gradients of functions in Uh(𝒯):=Pk+1(𝒯)H1(Ω)L02(Ω). Therefore, the uniqueness in (7.2) can be obtained in the following formulation: Find (ph,αh,wh)Xh(𝒯)×YN,k(𝒯)×Uh(𝒯) with

(ph,qh)L2(Ω)+(qh,Curlαh)L2(Ω)=(φ,qh)L2(Ω)for all qhXh(𝒯),(7.3a)(ph,Curlβh)L2(Ω)+(βh,wh)L2(Ω)=0for all βhYN,k(𝒯),(7.3b)(αh,vh)L2(Ω)=0for all vhUh(𝒯).(7.3c)

Note that Uh(𝒯) is the kernel of Curl:YN,k(𝒯)Pk(𝒯;3) and so (7.3) implies wh=0. 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 k=0, ph 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 Uh(𝒯)YN,k(𝒯) and CurlYN,k(𝒯)Xh(𝒯). This and the conformity of the method lead to the best-approximation result

p-phL2(Ω)+Curl(α-αh)L2(Ω)+(w-wh)L2(Ω)minqhXh(𝒯)p-qhL2(Ω)+minβhYN,k(𝒯)Curl(α-βh)L2(Ω)+minshUh(𝒯)(w-sh)L2(Ω).

Since w=wh=0, this is equivalent to

p-phL2(Ω)+Curl(α-αh)L2(Ω)minqhXh(𝒯)p-qhL2(Ω)+minβhYN,k(𝒯)Curl(α-βh)L2(Ω).

The following lemma states a projection property similar to Lemma 3.12 for the two-dimensional case. To this end, define

Zh(𝒯):={βhYN,k(𝒯)vhUh(𝒯):(βh,vh)L2(Ω)=0},Wh(𝒯):={qhXh(𝒯)βhZh(𝒯):(qh,Curlβh)L2(Ω)=0}.

Since Uh(𝒯) is the kernel of Curl:YN,k(𝒯)Xh(𝒯), it holds

CurlYN,k(𝒯)=CurlZh(𝒯).

This implies

Wh(𝒯)={qhXh(𝒯)βhYN,k(𝒯):(qh,Curlβh)L2(Ω)=0}.

(Projection Property)

Let qL2(Ω;R3) with (q,Curlβ)L2(Ω)=0 for all βH(Curl,Ω) (that means that q is a gradient of an H01(Ω) function). Then ΠXh(T)qWh(T). If T is an admissible refinement of T, then ΠXh(T)Wh(T)Wh(T).

Proof.

Since CurlYN,k(𝒯)Xh(𝒯) and YN,k(𝒯)H(Curl,Ω), 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

λ2(𝒯,T):=h𝒯CurlNCphL2(T)2+hTE(T)[phνE]EL2(E)2,μ2(T):=φ-ΠXh(𝒯)φL2(T)2

and (6.2), (6.3). Here, (T) denotes the faces of a tetrahedron T𝒯 and h𝒯P0(𝒯) denotes the piecewise constant mesh-size function defined by h𝒯|T:=hT:=meas3(T)1/3. The refinement of triangulations in Algorithm 1 is done by newest-vertex bisection [40]. Let 𝕋(N) denote the space of admissible triangulations with at most N tetrahedra more than 𝒯0. As in Section 6.1, define the seminorm

|(p,α,φ)|𝒜s:=supN0Nsinf𝒯𝕋(N)(p-ΠXh(𝒯)pL2(Ω)+infβ𝒯YN,k(𝒯)Curl(α-β𝒯)L2(Ω)+φ-ΠXh(𝒯)φL2(Ω)).

Assume that Assumption 6.4 holds. The following theorem states optimal convergence rates for Algorithm 1 for 3D.

(Optimal Convergence Rates of AFEM for 3D)

Let s>0. For 0<ρB<1 and sufficiently small 0<κ and 0<θ<1, Algorithm 1 computes sequences of triangulations (T)N and discrete solutions (p,α)N for the right-hand side φ of optimal rate of convergence in the sense that

(card(𝒯)-card(𝒯0))s(p-pL2(Ω)+Curl(α-α)L2(Ω))|(p,α,φ)|𝒜s.

The proof follows as in Section 6 from (A1)–(A4) and (B) from [16] and the efficiency of λ and μ. The proof of efficiency follows with the standard bubble function technique [43]. 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 h𝒯3|Th𝒯3|T/2 for all T𝒯𝒯. However, for (A3) quasi-orthogonality and (A4) discrete reliability, the interpolation operator of [39] cannot be applied directly to r𝒯YN,k(𝒯) as done in the proof of Theorem 6.8, because YN,k(𝒯)H1(Ω;3). This can be overcome by a quasi-interpolation based on a quasi-interpolation operator from [38] and a projection operator from [44]. Its properties are summarized in the following theorem.

(Quasi-Interpolation)

Let T be an admissible refinement of T and define

(𝒯,𝒯):={T𝒯K1𝒯𝒯,K2𝒯 with K1K2 and TK2}.

Let γTZh(T). Then there exist γTYN,k(T), ρH1(Ω), and ΦH1(Ω;R3) with

γ𝒯-γ𝒯=ρ+Φ,(γ𝒯-γ𝒯)|T=0for all T𝒯(𝒯,𝒯),h𝒯-1ΦL2(Ω)+ΦL2(Ω)Curlγ𝒯L2(Ω).

Proof.

This follows as in the proof of [44, Theorem 5.3] and with the ellipticity on the discrete kernel from [2, Proposition 4.6]. ∎

The differences between the proof of (A4) discrete reliability and the proof of Theorem 6.8 are outlined in the following. Let (p𝒯,α𝒯)Xh(𝒯)×Zh(𝒯) and (p𝒯,α𝒯)Xh(𝒯)×Zh(𝒯) denote the discrete solutions to (7.2). As in the proof of Theorem 6.8, let σ𝒯Wh(𝒯) and r𝒯Zh(𝒯) such that p𝒯-p𝒯=σ𝒯+Curlr𝒯. The first term of the right-hand side of

p𝒯-p𝒯2=(p𝒯-p𝒯,σ𝒯)L2(Ω)+(p𝒯-p𝒯,Curlr𝒯)L2(Ω)

is estimated as in the proof of Theorem 6.8, while for the second term, the quasi-interpolant r𝒯YN,k(𝒯) of r𝒯 with r𝒯-r𝒯=ρ+Φ for ρH1(Ω) and ΦH1(Ω;3) from Theorem 7.3 is employed. This yields

(p𝒯-p𝒯,Curlr𝒯)L2(Ω)=(p𝒯,Curl(r𝒯-r𝒯))L2(Ω)=(p𝒯,CurlΦ)L2(Ω).

A piecewise integration by parts and the arguments of the proof of Theorem 6.8 conclude the proof. The crucial point is that ΦH1(Ω;3) 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)) α-1Zh(𝒯-1)YN,k(𝒯M), there exists γMZh(𝒯M) with CurlγM=Curlα-1. Theorem 7.3 guarantees the existence of β-1YN,k(𝒯-1), ρH1(Ω) and ΦH1(Ω;3) with αM-γM-β-1=ρ+Φ. This implies in (6.12) that

(Curl(αM-α-1),p-1)L2(Ω)=(Curl(αM-γM-β-1),p-1)L2(Ω)=(CurlΦ,p-1)L2(Ω).

Since ΦH1(Ω;3) is smooth enough, a piecewise integration by parts and the arguments of the proof of Theorem 6.8 then prove

(Curl(αM-α-1),p-1)L2(Ω)(λ-1+μ-1)Curl(αM-α-1)L2(Ω).

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 k=0,1,2. Sections 8.18.3 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 θ=0.1 and κ=0.5 and ρ=0.75. The convergence history plots are logarithmically scaled and display the error p-phL2(Ω) against the number of degrees of freedom (ndof) of the linear system resulting from the Schur complement. The underlying L-shaped domain Ω:=(-1,1)2([0,1]×[-1,0]) with its initial triangulation is depicted in Figure b.

Red-refined triangle and initial mesh for the L-shaped domain.

(a)
(b)

8.1 L-Shaped Domain, I

The function u given in polar coordinates by

u(r,ϕ)=r2/3sin((2/3)ϕ)

is harmonic. For the following experiment we choose φ0 and uD:=gu with perturbation function gH2(Ω),

g(x):={0 if |x|1/2,16|x|4-64|x|3+88|x|2-48|x|+9 if 1/2|x|1,1 if |x|1,

such that g|Γ=1 for Γ:=Ω({0}×(-1,0)(0,1)×{0}). Since u|ΩΓ=0, it holds uD|Ω=u. Let

B1/2(0):={x2|x|<1/2}

denote the ball with radius 1/2 and midpoint (0,0). Since g|B1/2(0)=0 and uH2(ΩB1/2(0)), it holds uDH2(Ω).

For non-homogeneous Dirichlet data, the jump [ph]EτE is defined for boundary edges E, EΓD, with adjacent triangle T+ by

[ph]EτE:=ph|T+τE-uDτE.

The error estimator λ is then defined by (6.1)–(6.3). The local data error estimator contributions read

μ2(T):=(φ-uD)-Πk(φ-uD)L2(T)2.

The global error estimator μ is defined by (6.2) and (6.3).

The errors and error estimators for the approximation phPk(𝒯;2) of u for k=0,1,2 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 h2/3𝗇𝖽𝗈𝖿-1/3 for k=0,1,2. The adaptive refinement reproduces the optimal convergence rates of 𝗇𝖽𝗈𝖿-(k+1)/2 for k=0,1,2. Figure 3 depicts three meshes created by the adaptive algorithm for k=0, 1, and 2 with approximately 1000 degrees of freedom. The singularity at the re-entrant corner leads to a strong refinement towards (0,0), while the refinement for k=0,1 also reflects the behavior of the right-hand side, i.e., one also observes a moderate refinement on the circular ring {xΩ1/2|x|1}. The marking with respect to the data-approximation (μ2>κλ2 in Algorithm 1) is applied at the first 7 (resp. 5 and 10) levels for k=0 (resp. k=1 and k=2) and then at approximately every third level.

Errors and error estimators from Section 8.1.
Figure 2

Errors and error estimators from Section 8.1.

Adaptively refined triangulations for the experiment from Section 8.1.Adaptively refined triangulations for the experiment from Section 8.1.Adaptively refined triangulations for the experiment from Section 8.1.
Figure 3

Adaptively refined triangulations for the experiment from Section 8.1.

8.2 L-Shaped Domain, II

For f-1 and uD0 define φ(x,y):=(1/2)(x,y) with -divφ=f.

The error estimators are plotted against the degrees of freedom in Figure 4 for k=0,1,2. The error estimators show for k=0,1,2 a suboptimal convergence rate of h2/3𝗇𝖽𝗈𝖿-1/3 for uniform refinement. The adaptive Algorithm 1 recovers the optimal convergence rate of 𝗇𝖽𝗈𝖿-(k+1)/2. 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 φP1(Ω;2) implies that the data-approximation error estimator μ vanishes on all triangulations for k=1,2. For k=0, μ does not vanish, nevertheless, since μ2κλ2 for all , only the Dörfler marking is applied.

Error estimators for the experiment from Section 8.2.
Figure 4

Error estimators for the experiment from Section 8.2.

Adaptively refined triangulations for the experiment from Section 8.2.Adaptively refined triangulations for the experiment from Section 8.2.Adaptively refined triangulations for the experiment from Section 8.2.
Figure 5

Adaptively refined triangulations for the experiment from Section 8.2.

Errors and error estimators for the experiment with singular α
from Section 8.3.
Figure 6

Errors and error estimators for the experiment with singular α from Section 8.3.

Errors and error estimators for the experiment with singular α
from Section 8.3 and uniform refinement.
Figure 7

Errors and error estimators for the experiment with singular α from Section 8.3 and uniform refinement.

Adaptively refined triangulations for the experiment from Section 8.3.Adaptively refined triangulations for the experiment from Section 8.3.Adaptively refined triangulations for the experiment from Section 8.3.
Figure 8

Adaptively refined triangulations for the experiment from Section 8.3.

8.3 Singular α

This subsection is devoted to a numerical investigation of the dependence of the error p-phL2(Ω) on the regularity of α. The exact smooth solution uC(Ω) of

-Δu=2sin(πx)sin(πy) in Ωandu|ΓD=0

reads u(x,y)=sin(πx)sin(πy). Define φ=u+Curl(α~) with α~H1(Ω)H2(Ω) defined by

α~(r,ϕ)=r2/3sin(2ϕ/3).

Then φH(div,Ω) with -divφ=f.

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 k=1,2 is h2/3𝗇𝖽𝗈𝖿-1/3 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 k=0 and uniform mesh-refinement. The error p-phL2(Ω) and the error estimator λ2+μ2 show a convergence rate between h and h2/3, while Curl(α-αh)L2(Ω) converges with a rate of h2/3𝗇𝖽𝗈𝖿-1/3 due to the singularity of α. This numerical experiment suggests that the error p-phL2(Ω) does not depend on the regularity of α (at least in a preasymptotic regime). The triangle inequality implies Curl(α-αh)L2(Ω)p-phL2(Ω)+μ. This upper bound is also plotted in Figure 7.

Figure 8 depicts adaptively refined meshes for k=0,1,2 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 (μ2>κλ2 in Algorithm 1) is only applied at levels 1–5, 7, 12, and 18 for k=0. All other marking steps for k=0,1,2 use the Dörfler marking (μ2κλ2).

Acknowledgements

The author would like to thank Professor C. Carstensen for valuable discussions.

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About the article

Received: 2016-09-27

Accepted: 2016-09-29

Published Online: 2016-10-18

Published in Print: 2017-01-01


Citation Information: Computational Methods in Applied Mathematics, Volume 17, Issue 1, Pages 161–185, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2016-0031.

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