EQUILIBRATED RESIDUAL ERROR ESTIMATOR FOR EDGE ELEMENTS

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1 MAHMAICS O COMPUAION olume 77, Number 262, April 2008, Pages S (07) Article electronically published on November 20, 2007 QUILIBRAD RSIDUAL RROR SIMAOR OR DG LMNS DIRICH BRASS AND JOACHIM SCHÖBRL Abstract. Reliable a posteriori error estimates without generic constants can be obtained by a comparison of the finite element solution with a feasible function for the dual problem. A cheap computation of such functions via equilibration is well known for scalar equations of second order. We simplify and modify the equilibration such that it can be applied to the curl-curl equation and edge elements. he construction is more involved for edge elements since the equilibration has to be performed on subsets with different dimensions. or this reason, Raviart homas elements are extended in the spirit of distributions.. Introduction Recently, a posteriori error estimates without constants have attracted much interest; see Ainsworth and Oden [], Neittaanmäki and Repin [7], ejchodský [20], Luce and Wohlmuth [3] and also Ladevèze and Leguillon [2]. At first glance they look like estimators which use local Neumann problems as introduced by Bank and Weiser [5], but they are based on a comparison of primal and dual forms of the variational problems. ollowing Prager and Synge (949) in the special case of the Poisson equation u = f in Ω, one compares a finite element approximation u h H (Ω) and a function σ H(div) that satisfies the equilibrium condition div σ + f = 0. In principle, the latter can be obtained via mixed methods, but in practical computations a feasible function σ is constructed by an equilibration of u h. he equilibration can be done by solving local problems; see [, Chapter 6.4]. he solution of local problems by polynomials of sufficiently high order is avoided in [20] by the combination with a variant of the hypercircle method and in [3] by the introduction of a dual mesh. We will go a different way in order to avoid generic constants in the main term of the upper estimate. A small portion of the error that results from the data oscillation is estimated in the classical manner. So the local problems can be solved on finite-dimensional spaces, and no generic constant enters via approximation arguments. Moreover, the local problems are solved on patches around vertices of the mesh in order to avoid nonlocal auxiliary quantities. he procedure becomes transparent, and a generalization to other types Received by the editor July 26, 2006 and, in revised form, ebruary 20, Mathematics Subject Classification. Primary 65N30. Key words and phrases. A posteriori error estimates, Maxwell equations. he second author acknowledges support from the Austrian Science oundation W within project grant Start Y-92, hp-m: ast Solvers and Adaptivity. 65 c 2007 American Mathematical Society Reverts to public domain 28 years from publication

2 652 DIRICH BRASS AND JOACHIM SCHÖBRL of elliptic problems is now natural (although not trivial). In the 2D case there is even a simple geometrical interpretation of the resulting equilibration procedure. In the present paper we also establish a posteriori error estimators with similar properties for edge elements and the equations of magnetostatics. here is an analogue to the result of Prager and Synge although we have to deal with different Sobolev spaces and edge elements. he equilibration in H(curl), however, is more involved since the splitting of the residual currents into local divergence-free currents has to be done with more constraints. Moreover, the constraints refer to currents on geometrical objects with different dimensions. o overcome these obstacles we proceed as we have shown for the Poisson equation. We extend the Raviart homas elements and Nédélec elements to finite element spaces such that the differential operators curl and div act on distributions. We show that the differential operators and the extended spaces still form exact sequences. he sequences generalize the de Rham sequences and the discrete analogs that were frequently used in the last years for constructing and understanding new finite element spaces [2, 3, ]. In Section 2 we write the equilibration procedure for the (scalar) Poisson equation in our setting in order to make the reader familiar with the modifications for avoiding generic constants. he resulting local problems will differ from those in the literature. he distributional Raviart homas elements and the corresponding Nédélec elements are introduced in Section 3. Details are provided for the 2-dimensional case while the discussion of the 3-dimensional case is treated more briefly. Section 4 contains the application to a posteriori error estimators for the curl-curl equation. he Sobolev spaces H (Ω), H0 (Ω), H(div, Ω) and H(curl, Ω) are defined as usual. he specification of the domain will often be suppressed when there is no danger of ambiguity. 2. quilibrated residual error estimates for scalar equations In this section we consider the scalar equation (2.) u = f on a polygonal domain Ω in 2-space or 3-space. Moreover, let u = 0 on a nonempty subset Γ D Ω and u/ n =0onΓ N := Ω\Γ D. he a posteriori error estimates in [, 2, 7, 5, 3, 20] are related to a result of Prager and Synge [8] although their presentations are very different. We provide ideas for achieving estimates without generic constants and with simpler local problems. or convenience, we restrict ourselves to the Poisson equation. he generalization to equations with piecewise constant coefficients will be clear from the considerations in Section 4. he finite element solutions of (2.) are determined on a triangulation of Ω into triangles or tetrahedra, Ω =.Let M k := {v L 2 (Ω); v P k }, M k 0 := M k C 0 (Ω) be the sets of polynomial Lagrange finite elements on the triangulation above, and let u h h := M 0 (with the essential boundary conditions incorporated) be the finite element solution for linear elements, i.e., (2.2) ( u h, v) =(f,v) for v h.

3 QUILIBRAD RROR SIMAOR OR DG LMNS 653 he distribution (2.3) f h := u h is a functional on H (Ω) and evaluates to (2.4) f h,v = ( u h, v) = =: f h, v. { } u h u h v + n v Here runs over the faces of the elements (and over the edges in the 2D case, resp.), and f h, := [ u h n ]:= u h,l n l + u h,r n r. In particular, the right-hand side of (2.4) is understood as the face contributions of the divergence of u h. Since we treat the d-dimensional case for d =2andd = 3 simultaneously, we use the letter for (d )-dimensional simplices. In particular, (2.4) describes the face contribution of the divergence of u h. In general, contributions of tetrahedral/triangular elements, faces, edges and vertices are distinguished by the labels,,, and, respectively. 2.. he theorem of Prager and Synge. heorem (heorem of Prager and Synge). Let σ H(div), σ n =0on Γ N while v H (Ω), v =0on Γ D and assume that (2.5) div σ + f =0. urthermore, let u be the solution of the Poisson equation (2.). hen, (2.6) u v 2 + u σ 2 = v σ 2. A proof can be found in [8, 8]; see also heorem 0. We are looking for a cheap construction of a function σ that satisfies (2.5). irst we assume that f is constant on each element. hen there is such a function σ in the Raviart homas space R : R := {τ [L 2 (Ω)] d ; τ = a + b x, a R d,b R }, R := R H(div). If we solve the original equation (2.) by the mixed method with the Raviart homas element [8, pp. 48, 8], then we would yield the function σ R with (2.5) for which u h σ is minimal. Indeed, it follows from (2.6) that this is equivalent to the minimization of u σ, and here the minimum is attained at the solution of the mixed method of Raviart homas. his procedure, however, would be too expensive for computing a posteriori error estimates. We rather construct a function σ satisfying (2.5) from the given finite element solution u h by a local procedure usually called equilibration. Weperformtheconstruction for the difference σ u h =: σ. Obviously, σ belongs to the broken Raviart homas space R defined above. So we proceed in R and not on

4 654 DIRICH BRASS AND JOACHIM SCHÖBRL the continuous level. he conditions σ + u h H(div) and div(σ + u h )= f are rewritten as (2.7) div σ = f in, [ σ n ] = [ u h n] on quilibration. Given a vertex of the mesh, we assign to it the patch ω := {, }. he correction σ will be constructed from the solutions σ ω of local problems on the patches: (2.8) σ = σ ω. Here runs over all vertices of the triangulation, and supp σ ω ω. We recall that f is assumed to be constant on each element. Let be a node of the mesh and ψ be the linear nodal function with ψ ( )=andψ (x) =0for x Ω\ω. rom the characterization (2.2) of u h as a finite element solution and by partial integration we obtain fψ = u h ψ = [ uh ] ψ. n ω ω ω Since ψ is piecewise linear, we have ψ dx = d dx = d, andweobtain the central relation of this section: (2.9) fψ = f h,. d ω ω Now, we fix the functions of the decomposition (2.8) by div σ ω = fψ in ω, (2.0) [σ ω n] = d [ u h n] on ω, σ ω n = 0 on ω. Remark 2. here are some modifications at the vertices on the boundary of Ω. If ω Ω Γ N, there is no change apart from the geometry. If Γ D,then there is no test function associated with the vertex, and (2.9) does not hold for the vertex. In this case, however, there is no boundary condition on ω Γ D when we construct σ ω. here is not a problem. hus we will ignore adaptations at boundaries in the sequel. he existence of solutions of (2.0) follows from the following lemma when it is applied to the patches ω. he assumption in the lemma is guaranteed by (2.9). Since ψ = and each face has d vertices, indeed, the sum (2.8) yields a solution of (2.7). Lemma 3. Let ω = Rd, d =2or 3, be simply connected, and let = \ ω be a decomposition of the interelement boundaries. If the distribution g, (2.) g, v := g v + g v

5 QUILIBRAD RROR SIMAOR OR DG LMNS 655 with piecewise constant functions g, g satisfies g, =0, then there exists σ R such that (2.2) σ n = 0 on ω, div σ = g in ω, [σ n] = g on ω. Moreover, there exists a constant c depending only on the shape parameter of the mesh such that ( σ 2 0 c h 2 g 2 L 2 ( ) + ) h g 2 L 2 ( ). Proof. irst we reduce the given equations to a problem without face terms. We choose σ R by setting σ n = 2 g at internal interfaces and σ n =0on ω. hus, the face contributions of div σ coincide with the face contributions of g, and the difference is the regular function. Moreover, by Gauss theorem div σ, = div σ [σ n] = 0. Hence, g div σ M 0 L 2 (Ω) and g div σ, =0. rom Remark 2. in [3] we know that the sequence (2.3) R 0,0 div M 0 R is exact, where R 0,0 := {τ R, τ n =0on ω} and the second mapping is defined by :g g,. husthereexistsσ 0 R such that div σ 0 = g div σ in ω, σ 0 n = 0 on ω. Setting σ := σ 0 + σ we obtain a solution of (2.2). he stability estimate will be proven in Section 3.4 for more general cases. here is also a constructive proof if the analysis is restricted to the 2-dimensional case as in [8, p. 83]. he first result of the lemma can be understood as an extension of (2.3) being an exact sequence. Details will be given in the next section Data oscillation. ventually, we want to abandon the assumption that f is piecewise constant. Let f be the L 2 -projection of f onto piecewise constant functions. Since fψ = f, the preceding investigation applies to the error if the right-hand side of (2.) is replaced by f. Now the difference between the solutions for f and f can be bounded by (2.4) ch f f. his effect of the data oscillation is well known [8, p. 74]. We emphasize that the constant c depends on the shape of the elements, but it does not depend on the domain Ω. Since (2.4) is a term of higher order, we can admit a generic constant here.

6 656 DIRICH BRASS AND JOACHIM SCHÖBRL 2.4. fficiency. By construction, the error estimate (u u h ) σ + ch f f is reliable. By Lemma 3, σ can be bounded by the terms h f and h /2 f f h,, i.e., by the ingredients of the well-known residual error estimator. hus σ is bounded by a multiple of that estimator. Since the residual error estimator is efficient, the same holds for the estimates determined by equilibration. he results of this section are summarized for the Poisson equation as follows. heorem 4. or each node there exists a broken R-function σ ω with support in ω and satisfying (2.0). Choose σ ω with (quasi-) minimal L 2 -norm, and let σ := σ ω. hen we have the a posteriori error estimate (2.5) c 0 σ ch f f (u u h ) σ + ch f f. 3. Distributional finite element de Rham sequences In the treatment of the scalar equation we already encountered distributional finite elements. In this section, we introduce and study exact sequences of finite elements which contain more distributional terms and are suitable for the equation of magnetostatics. We start with the two-dimensional case and continue with three-dimensional finite elements. Let Ω be a simply connected domain in R 2. In2D,wewritecurlforthedifferential operator ( y, x ). hen, the de Rham sequence (3.) R H curl H(div) div L 2 0 is an exact sequence [2]. his means that the operator curl has a trivial kernel in H /R; the kernel {σ H(div) : div σ =0} of the operator div is exactly the range of the operator curl; the range of the operator div is exactly L 2. An analogous property holds for the spaces with zero boundary conditions H0 and H 0 (div) := {σ H(div) : σ n =0on Ω}: (3.2) 0 H0 curl H 0 (div) div L 2 R 0. As usual, the space L 2,0 := {f L 2 : f =0} of functions with zero mean values Ω is identified with L 2 /R. We focus on sequences without boundary conditions, i.e., on sequences of type (3.) in the following introductory discussion although we will deal later also with generalizations of (3.2). Note that we find the right-hand part of the last exact sequence on the discrete level in (2.3). 3.. irst distributional triangular elements. he exact sequence property is inherited on the discrete level when we choose piecewise linear and continuous Lagrangian elements M 0 for modeling H, the Raviart homas elements R for H(div), and piecewise constant, noncontinuous elements M 0 for L 2, [7, p. 75]: (3.3) R M 0 curl R div M 0 0.

7 QUILIBRAD RROR SIMAOR OR DG LMNS 657 igure. Classical finite element spaces in the sequence (3.3) Let u M 0, σ R,andf M0. heir natural degrees of freedom are nodal values û := u( ), edge integrals of the normal components σ := σ n, and element integrals f := f, respectively. Note that an orientation is associated with each edge for defining the normal components of R functions. Here and below, symbols with a hat refer to the integral over the geometrical object specified by the superscript. he representation of the differential operators with respect to these degrees of freedom depends only on the element topology and is independent of the shape of the elements. In terms of degrees of freedom we find σ =curlu as σ = û, û,2, where, and,2 are the two vertices of the edge, ordered consistently with the previously defined normal vector. Similarly, the expression f =divσ reads as f = ± σ, where the sign depends on the orientation of the normal vector. Specifically, the sign is positive for normal vectors pointing to the outside of the triangle. An element f in M 0 generates the regular distribution f,v = f v. or our purposes we introduce the space M 0 3 of distributions involving element, edge, and vertex terms: (3.4) f,v = f v + f v + f v( ). he functions f and f are constant on each triangle and edge, respectively. he subspace of distributions of the form (3.4) with vanishing vertex terms is denoted as M 0 2. irst, we restrict ourselves to those distributions with element and vertex terms, i.e., with f =0forall ; see also (2.). In this context we recall the extension of the Raviart homas space to the broken Raviart homas space and obtain the first distributional de Rham sequence (3.5) R M curl div 0 R M he sequence (3.5) is well defined. his is clear for the curl operator. o verify it

8 658 DIRICH BRASS AND JOACHIM SCHÖBRL igure 2. Distributional finite element spaces in the sequence (3.5). he middle lines of the edges represent the edge terms in (3.4) for the divergence, let σ R, and define f =divσ in the distributional sense by f,v := σ, v for v C0. Integration by parts leads to f,v = σ v = div σv σ nv = div σv σ n v. : Here the normal vectors are defined element by element and as usual in the outgoing direction; cf. igure 2. hus the image div σ belongs to M 0 2, andtherelation f =divσ evaluates to two relations (3.6) f =div σ and f = σ n. : Since σ R is determined by the fluxes on each side of the edges, we have in terms of degrees of freedom f = σ and f = σ. heorem 5. he sequence (3.5) is exact. : Proof. We recall that the classical sequence (3.3) is exact [2, 7, 9,, 4]. Due to (3.6) the properties σ R and div σ = 0 imply that σ R. Hence, the divergence is defined as usual in H(div) and vanishes. Now the exactness of (3.3) guarantees that Raviart homas elements with vanishing divergence are curls of functions in M 0. he surjectivity of the divergence onto M 0 2 is also obtained from the exactness (3.3) by similar arguments as for the reduction in the proof of Lemma 3 (cf. also the reduction in the proof of the next theorem).

9 QUILIBRAD RROR SIMAOR OR DG LMNS 659 f f f igure 3. Distributional finite element spaces in the sequence (3.7). he arrows along the edges represent the edge and the vertex terms in (3.8) 3.2. Second distributional triangular elements. or the treatment of the curlcurl equations, we need another extension of the sequence. Distributional elements on edges are added to the finite elements that model H(div), and the entire set M 0 3 of distributions of the form (3.4) enters into the theory. he associated sequence will be called the second distributional de Rham sequence: (3.7) R M curl R 2 div M he space M consists of piecewise linear and noncontinuous finite elements. he degrees of freedom are the values û at the three vertices of each triangle; seeigure3. he corresponding Raviart homas distributions are of the form σ, v = σ v + σ v, where σ = a + b x, andσ =(a + bx) τ are D Raviart homas elements mapped to the edge where τ is a tangential vector. he degrees of freedom are (3.8) σ = σ n and σ = σ ( ) n. Here n is the vector pointing outwards at the vertex of an edge. he representation of the operation f =divσ in terms of the degrees of freedom is f = σ, (3.9) f = σ σ, : f = σ. : Remark 6. In contrast to the previous case, div σ = 0 is now possible for elements σ that are not in the classical Raviart homas space R. he distributional parts of div σ may add to zero in (3.9). Nevertheless, there is a geometrical understanding. We may blow up the edges to slim rectangles (in an exploded mesh);see igure 4. If

10 660 DIRICH BRASS AND JOACHIM SCHÖBRL igure 4. Slim rectangles in the sense of Remark 6 the divergence vanishes in the distributional sense, the total flow over the boundary of a slim rectangle (and not only over the boundary of the triangles) is zero. he imagination with the slim rectangles has another advantage. he (classical) Raviart homas elements in 2-space are given by the fluxes on the edges. Now all the degrees of freedom of the distributional Raviart homas elements are fluxes on edges, i.e., they live on -dimensional objects. he terms on the right-hand side of (3.9) are fluxes over boundaries of triangles, slim rectangles, or the central area in igure 4. he differential operation σ =curlu reads as (3.0) σ σ = û, û,2, = û û 2, where is the left and 2 is the right triangle when looking into the direction of n v. heorem 7. he second distributional de Rham sequence is exact. Proof. We start with proving that the operator div is a mapping onto M 0 3. Given f M 0 3,wefirstchooseσ2 such that the vertex terms of div σ 2 coincide with the vertex terms of f. o this end we set where N σ 2 = N f, is the number of edges sharing the vertex. rom (3.9) 3 we know that f div σ 2 M 2. By the first distributional exact sequence, there exists σ R such that div σ = f div σ 2. Hence, div(σ + σ 2 )=f. his proves that the divergence operator is surjective and proves the exactness of the second operator. Next, consider σ R 2 with div σ = 0. We construct a function u 2 M such that the edge terms of curl u 2 coincide with the edge terms of the given σ.

11 QUILIBRAD RROR SIMAOR OR DG LMNS 66 his is done by a local construction for each vertex. Given a vertex, we conclude from the vertex part of div σ that σ =0. : Now, enumerate the triangles sharing the vertex from to N. numerate the edges such that i is between triangle i and i+ mod N.Weset u 2 ( )=0 and u 2 i+ ( )=u 2 i ( )+ σ i. Since the vertex currents sum to 0, it follows that u 2 ( )=u 2 N ( )+ σ N By construction, the edge terms of curl u 2 coincide with the edge terms of σ. hus, the difference satisfies σ curl u 2 R and div(σ curl u 2 )=0. We know from heorem 5 that there exists u M 0 such that curl u = σ curl u 2, and u = u + u 2 is the desired function in M Distributional tetrahedral elements. he three-dimensional de Rham sequence contains an additional space. In the case of zero boundary conditions it reads (3.) 0 H0 grad H 0 (curl) curl H 0 (div) div L 2 R 0. he canonical lowest order finite elements inherit the exact sequence property (3.2) 0 M grad 0 Nd curl R div M 0 R 0; see [4]. Here, Nd consists of the lowest order Nédélec elements. We define the space M 0 4 of scalar distributions of the form (3.3) f,v = f v + f v + f v + f v( ), where f, f,andf are piecewise constant functions on tetrahedra, faces, and edges, respectively. he f are real numbers. he subspaces M 0 M 0 2 M 0 3 of lower distributional orders are defined to contain only element terms, element and face terms, and element, face, and edge terms, respectively.. Moreover, we define the space R 3 of H(div) distributions of the form (3.4) σ, v = σ v + σ v + σ v, where σ, σ,andσ are in the Raviart homas element space on tetrahedra, triangular faces in 3D space, and edges in 3D. he degrees of freedom are the normal fluxes through the boundary. Specifically, we take the normal flux σ of σ through the face, the normal flux σ of σ through the edge, and the flux σ of σ into the vertex of. he degrees σ of a face flux are depicted in igure 5.

12 662 DIRICH BRASS AND JOACHIM SCHÖBRL igure 5. Raviart homas face distribution Only element and face distributions are required for modeling the space H(curl), (3.5) H, v = H v + H v. hese distributions generate the space Nd 2,andNd is the subspace with vanishing face terms. he finite element functions are spanned in each tetrahedron and triangle by Nédélec shape functions. heir degrees of freedom are the tangential components along the tetrahedral and triangular edges. Note that there are jumps of the tangential components of a distributional Nédélec function between the tetrahedra, and individual values of the components are given on common edges. Now we are ready to formulate three sequences for distributional finite element spaces in R 3. hey differ by the order of the distributions. We focus on the spaces with boundary conditions (but the sequences for the versions without boundary conditions are also exact): (3.6) (3.7) 0 M 0 0 M 0 grad Nd 0 grad Nd curl R curl R 2 div M 0 2 div M 0 3 R 0, R 0, grad Nd 2 R 0. 0 M curl div (3.8) R 3 M 0 4 he first sequence (3.6) was already used for the construction of the equilibrated fluxes for the scalar equation in the previous section. he second sequence (3.7) will be used to construct the equilibrated magnetic fields for the curl-curl equation. he third sequence is formulated only for completeness. Lemma 8. he sequences (3.6), (3.7), and(3.8) are exact. div Proof. We start with the first sequence. he exactness of R M 0 2 R was already proven in Lemma 3. Since div σ =0forσ R implies that σ R,

13 QUILIBRAD RROR SIMAOR OR DG LMNS 663 the exactness of the rest of the sequence follows from the exactness of the standard finite element sequence. We continue with the sequence (3.7). Given f M 0 3 that contains element, face, and edge terms, we construct a σ R 2 such that div σ = f. We define the edge degrees of freedom for an auxiliary σ by σ := f, N where N is the number of faces sharing the edge. hus, the edge terms of div σ areequaltotheedgetermsoff, and thus f div σ M 0 2. he first distributional sequence yields the existence of a σ 2 R satisfying div σ 2 = f div σ,andwe have div(σ + σ 2 )=f. We turn to the middle part of (3.7). Given σ R 2 with div σ =0,weconstruct a function H Nd such that curl H = σ. rom the edge part of the divergence it follows that : σ =0. Wefixanedge and enumerate the tetrahedra and faces around the edges such that face i is between i and i+. Also here, element indices are taken modulo N. We define an H by Ĥ := 0 and Ĥ i+ := Ĥ i + σi. Since div σ = 0, we end up with H N+ = 0 after a complete cycle. he residual σ curl H is divergence free, and it is contained in R. We apply the first distributional sequence to ensure the existence of an H 2 Nd 0 such that curl(h + H 2 )=σ. o complete the second part, we pick an H Nd such that curl H =0. By definition H L 2 holdsaswellascurlh =0 L 2 ; thus H H(curl). his implies that the tangential components of H are continuous, i.e., H Nd. Bythe exactness of the standard sequence, there exists a ϕ M 0 such that grad ϕ = H. We skip the proof of the third sequence, since it follows the same lines, and it was added only for completeness Stability of inverses. or f M 0 4, σ R 3, H Nd 2,andu M we define the mesh-dependent norms f 2 0,h := f 2 L 2 ( ) + h f 2 L 2 ( ) + h 2 f 2 L 2 () + h 3 f 2, σ 2 0,h := σ 2 L 2 ( ) + h σ 2 L 2 ( ) + h 2 σ 2 L 2 (), H 2 0,h := H 2 L 2 ( ) + h H 2 L 2 ( ), u 2 0,h := u 2 L 2 ( ).

14 664 DIRICH BRASS AND JOACHIM SCHÖBRL Lemma 9. he right inverses of the differential operators constructed above satisfy the norm estimates σ 0,h ch f 0,h, where div σ = f, H 0,h ch σ 0,h, where curl H = σ, u 0,h ch H 0,h, where u = H. Proof. By transformation to the reference element (using standard, covariant, or the Piola transformation), { one easily proves that f 2 0,h h 3 ) 2 ) 2 ) 2 ( ) } 2 ( f + ( f + ( f + f, { σ 2 0,h h ( ) 2 ( ) σ + 2 ( ) } σ + 2 σ, { H 2 0,h h ) 2 ) } 2 (Ĥ + (Ĥ, { u 2 0,h h 3 (û ) } 2. Define f as the vector containing all degrees of freedom. he relation div σ = f can be written as a singular, but consistent matrix equation for the coefficient vectors B div σ = f, where the matrix B div is defined according to (3.9). All matrix elements are either +,, or 0. he matrix depends only on the topology of the mesh. Assuming a patch of shape regular elements, there is only a finite number of possible topologies, and thus there exists a common constant c such that σ R n c f R n. ogether with the norm equivalences the statement follows. Similar arguments on matrices with entries +,, and 0 in this context can be found in [3]. 4. quilibration in H(curl) We consider the curl-curl equation for the vector potential: ind u H(curl) such that (µ curl u, curl v) =(j, v) for v H(curl). he given current density j is supposed to be divergence free. Moreover, we assume that j is element-wise constant. hus, j can be represented by means of Raviart homas functions. We are interested in a posteriori error estimates of the finite element discretization u h with Nédélec elements of lowest order, (µ curl u h, curl v) =(j, v) for v Nd. he magnetic field H defined as (4.) H := µ curl u

15 QUILIBRAD RROR SIMAOR OR DG LMNS 665 satisfies Ampère s law (4.2) curl H = j. he magnetic field H h obtained from the finite element discretization, H h := µ curl u h, leads in general to a different current density (4.3) j h := curl H h. or piecewise linear vector potentials u h, the magnetic flux H h is piecewise constant, and the discrete curl, i.e. j h, is a face-based R distribution. 4.. An equation of Prager Synge type. he following result will be the basis of the error estimate. It is the analogue to heorem. heorem 0. Assume that v H(curl) satisfies the boundary conditions and that H H(curl) satisfies Ampère s law curl H = j. hen (4.4) µ /2 curl(u v) µ /2 (H H) 2 0 = µ /2 (curl v µ H) 2 0. Proof. Integration by parts yields the orthogonality relation curl(u v)(h H) Ω = (u v)curl(h H)+ [(u v) n] (H H) Ω Ω = (u v)(j j) =0. Ω By applying the binomial formula to µ /2 curl(u v)+µ /2 ( H H) and noting that µh =curlu we obtain (4.4). he lemma provides error estimates for curl u, and the estimate is independent of the gauge. he lemma above will be applied to v := u h. In order to achieve a good candidate for H we have to solve curl( H H h )=j j h. or this reason we are going to construct a correction H such that j j h =curlh. Again, we construct H ω locally on the vertex patch ω such that we obtain a decomposition H = H ω. he construction will be independent of the material parameter µ, and µ will enter only at the final end when heorem 0 will be applied. In particular, the coefficient µ may be piecewise constant on the mesh.

16 666 DIRICH BRASS AND JOACHIM SCHÖBRL 4.2. hediscretecurrent.he distribution j h is evaluated by using partial integration and recalling that H h is piecewise constant: j h,v = curl H h,v =(H h, curl v) = curl H h vdx+ [H h n] vds = ([H h n],v). he discrete current distributions are (4.5) j h, =[H h n]. Both currents, the prescribed current j as well as the discrete current j h can be represented by distributional Raviart homas elements of order. Both currents are divergence free. We utilize the properties of the Galerkin orthogonality, namely (4.6) j jh,ϕ =0 for each Nédélec basis function ϕ associated with the generic edge. Let and 2 be its two vertices. Given an edge of an element, the basis function can be expressed on the simplex in terms of the barycentric coordinates ϕ = λ λ 2 λ 2 λ ; see [4, (5.47)]. We recall that j as well as λ i is constant on the element and evaluate the contribution of j on an element sharing the edge : j ϕ = j (λ λ 2 λ 2 λ ) = (j λ 2 ) λ (j λ ) λ 2 = { } j λ2 j λ. 4 Now, observe that λ i is proportional to the normal vector on the face i that lies opposite to vertex i, and the factor is the inverse of the height of the element over the face i : λ i = h i n i = i 3 n i. hus, the element contributions evaluate to j ϕ = { } j n 2 j n 2 2 = { } j n j n = 2 } (4.7) {ĵ ĵ Note that the fluxes through element faces are the degrees of freedom of the Raviart homas elements. Similarly, the contribution of j h on a face is determined by j h, ϕ = { } j h n j h n 6 2 = 2 } {ĵh, ĵh, 6

17 QUILIBRAD RROR SIMAOR OR DG LMNS 667, /6 /6,2 /2 2 /6 /6 /2 igure 6. actors in relation (4.0) referring to an edge and adjacent triangles. he edge terms refer to j and the vertex terms to j h where, 2 are the endpoints of, and, 2 are the edges of the face which lie opposite to the vertices above. he normal vectors to the edges refer to the plane and are vectors in. he integrals above are inserted in (4.6) to derive a relation between the original and the discrete current: { 2 j n } j n,,2 : (4.8) = { 6 j h n j h n},,2 or (4.9) 2 : : {ĵ, ĵ,2 } 6 : {ĵh,, ĵh,, } = quilibration in 2D. he basic relation for the 2D model that corresponds to (4.9) can be established in the same way:,,2 } 2 } (4.0) {ĵ ĵ + {ĵh, ĵh, = : As above,, 2 are the endpoints of the edge under consideration, and,,,2 are the edges of the triangle which lie opposite to them; see igure 6. (he sign of the second term in (4.0) differs from that in (4.9), since, 2 refer directly to the points and not to objects opposite to them.) We proceed with the 2D case and are going to decompose the residual current into local, divergence free currents, i.e., j j h = j ω. We consider a generic node and the patch ω := {, }. Let be a triangle in ω and be an edge of the triangle sharing the vertex. he edge of

18 668 DIRICH BRASS AND JOACHIM SCHÖBRL O,O,P igure 7. Some notation for the local current j ω the triangle opposite to is denoted as,o. It is located on the boundary of ω. he third edge is denoted as,p.wedefinethelocalcurrentj ω on by ĵ ω, := + 6 (ĵ,o,p ĵ ), 2ĵ P ĵ ω, :=,P + 6 (ĵ,o (4.) ĵ ), 2ĵ O ĵ ω, := 0. Obviously, the setting is symmetric with respect to the two edges that share the vertex, but the representation with respect to a given edge will be more useful in the sequel. Moreover, the flow is fixed such that the flow on the boundary of ω is zero. Next, let be an edge in the patch ω that connects with a point O on ω. he vertex distributional parts are now fixed and evaluated from the fluxes on via (4.2) ĵ ω, := ĵ h,, O ĵ ω, := 0. By definition, this current also has zero flow on ω. Lemma. If j ω is defined by (4.) and (4.2), thendiv j ω =0and we have a decomposition (4.3) j j h = j ω. Proof. Let be a triangle with edge whose endpoints are and 2.Whenwe sum over all patches, only the patches with centers and 2 contribute to the sum of ĵ ω,. Recalling (4.) we have ĵ ω, = ĵ ω, + ĵω2, = 2ĵ + = ĵ 2ĵ

19 QUILIBRAD RROR SIMAOR OR DG LMNS 669 since the terms with the factor /6 in (4.) cancel in the sum. Moreover, only the patch with center contributes to the sum of ĵ ω,, Hence, = ĵω =, ĵh,. ĵ ω, he last two equations show that (4.3) holds. We consider now the divergence of j ω and do it recalling (3.9). By adding the terms in (4.) it follows that div,p,o j ω = ĵ ω, = ĵω, + ĵω, + ĵ ω, : = + 2ĵ 6 (ĵ,o,p ĵ ) +,P + 6 (ĵ,o ĵ )+0 2ĵ =,O,P } {ĵ + ĵ + ĵ 3 = j n = (4.4) div j = We obtain the edge terms from (4.) and (4.2): div,o j ω = ĵω, + ĵω, (4.5) = ĵ h, +0 : { 2ĵ : ĵ ω, + 6(ĵ,O j,p )}. Since the normal components of j are continuous, we have : 2 ĵ O rom div j h = 0 it follows that ĵ h, + ĵ h, = 0, and we continue with div O ),O j ω = (ĵh, ĵ h, (ĵ,p ) j = : =0. Here we applied the Galerkin equation (4.0) to = and to 2 = O. inally, the vertex terms are given by the flow into the center of the patch. rom the definition (4.2) we have (4.6) div j ω = ĵ ω, = j h, = div j h =0. : : his concludes the proof of div j ω =0. We note that (4.6) can be obtained from (4.4) and (4.5) by virtue of arguments in the spirit of Remark 6. Since the current vanishes on ω and the divergence on the triangles and the slim rectangles is zero, the total flux into must also be zero. (his argument is also helpful in the 3-dimensional case.) Since j ω is in R 2 with vanishing boundary values, we can apply the second distributional de Rham sequence to find an H ω in the scalar noncontinuous P space M with vanishing boundary values such that curl H ω = j ω. Recalling (3.0) we see that H ω is easily determined.

20 670 DIRICH BRASS AND JOACHIM SCHÖBRL 4.4. quilibration in 3D. We consider the construction in the 3-dimensional case very briefly. We construct the local current on the patch around a generic vertex. Regard a tetrahedron,aface andanedge such that. Let,O be the face opposite to,let,p be the face containing and opposite to, andlet,q be the remaining face containing. Similarly, let,o be the edge of the face opposite to and,p the edge of containing the vertex. he face terms on a tetrahedron depend only on j in ; cf. (4.). We set (4.7) ĵ ω, := +,O,P,Q } {ĵ + ĵ. 3ĵ 2ĵ 24 By symmetry, this defines also the fluxes through,p and,q. Moreover, the flux on the boundary of the patch is set to zero: ĵ ω,,o := 0. Contrary to the 2D case, the fluxes through faces depend not only on fluxes in faces, but involve also element terms. We set (4.8) ĵ ω, := { 2ĵh, + : + 6ĵh,,O 24 6ĵh, },O,P {ĵ ĵ.,p } Again, fluxes through the outer face are set to zero, i.e. ĵ ω,,o =0. Lemma 2. his is a local, divergence free decomposition of the residual, i.e., j j h = j ω and div j ω =0. We abandon the proof that proceeds along the lines of the proof of Lemma. he results of this section are summarized for the Maxwell equation in 3-space as follows. heorem 3. or each node there exists a broken Nédélec function H ω support in ω such that with curl H ω = j ω holds in the distributional sense, where j ω is defined by (4.7) and (4.8). Choose H ω with (quasi-) minimal L 2 -norm, and let H := H ω. hen the postprocessed magnetic flux H = µ curl u h + H satisfies Ampère s law curl H = j, and we have the a posteriori error estimate (4.9) c 0 µ /2 H µ /2 curl(u u h ) µ /2 H. Proof. he reliability follows from Lemma 2, the exactness of the second distributional de Rham sequence (3.7), and heorem 0. he efficiency estimate follows from the stability of the right inverse, Lemma 9, and the efficiency of the residual error estimator analyzed in [6].

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22 672 DIRICH BRASS AND JOACHIM SCHÖBRL aculty of Mathematics, Ruhr-University, D Bochum, Germany -mail address: Dietrich.Braess@rub.de Center for Computational ngineering Science, RWH Aachen University, D Aachen, Germany -mail address: joachim.schoeberl@mathcces.rwth-aachen.de URL: