A NOTE ON CONSTANT-FREE A POSTERIORI ERROR ESTIMATES

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1 A NOTE ON CONSTANT-FREE A POSTERIORI ERROR ESTIMATES R. VERFÜRTH Abstract. In this note we look at constant-free a posteriori error estimates from a different perspective. We show that they can be interpreted as an alternative way of expressing the residual of a finite element approximation and thus fit into the same framework as other a posteriori error estimates such as residual error indicators. Our approach also reveals that, when applied to singularly perturbed reaction-diffusion or convection-diffusion problems, constant-free a posteriori error estimates will not be fully robust unless extra measures are taken. 1. Introduction Nowadays a posteriori error estimates are an indispensable tool for the efficient numerical solution of partial differential equations. They typically yield upper and lower bounds of the form c 1 indicator error c indicator where the error is measured in a suitable norm and where the evaluation of the indicator should be cheap and should require only the knowledge of the given data of the differential equation and of its computed approximate solution. The upper and lower bounds are often referred to as reliability and efficiency, respectively. The product c c determines the quality of the error indicator and can be viewed as a sort of condition number. Of course it should be independent of any meshsize. In many situations it should also be independent of parameters inherent in the differential equation such as the size of reaction or convection terms relative to the diffusion. This property is often referred to as robustness and is indispensable for the efficient approximation of singularly perturbed problems. There is a large variety of error indicators such as, e.g. residual estimates, solution of discrete local auxiliary problems, hierarchical estimates, averaging techniques, equilibrated residuals. One can prove that all these indicators are equivalent in the sense that they can be Date: February 16, Mathematics Subject Classification. 65N30, 65N15, 65J10. ey words and phrases. Constant-free a posteriori error estimates, guaranteed upper bounds, residuals, dual norms, robustness for singularly perturbed problems, H(div)-lifting. 1

2 2 R. VERFÜRTH bounded from above and below by constant multiples of the residual estimates, cf. e.g. [18, Chap. I]. The basic ingredients for proving the reliability and efficiency of the residual estimates are: the equivalence of a norm of the error e and of the corresponding dual norm of the residual R in the sense of c 1 R e c R, an L 2 -representation of the residual, the Galerkin orthogonality of the discrete solution, local error estimates for suitable quasi-interpolation operators, inverse estimates for suitable local cut-off functions. The constants c and c which appear in the first step only depend on the differential equation and the chosen norm. In particular one has c c = 1 when using an energy norm. For more complex problems such as convection-dominated ones, c c still is of moderate size [20]. The methods listed above differ in the way how they bound the dual norm of the residual. Recently a different class of so called constant-free error indicators has become popular (cf. [4, 14] and the literature cited there for an overview). Here, the term constant-free means that these indicators always yield c = 1. Hence, they have the advantage that their reliability does not depend on a hidden constant which is not known explicitly or which is difficult to estimate sharply. But, it should be stressed that their condition, i.e. the product c c, is not 1 since the efficiency still depends on inverse estimates. In particular it is by no ways evident that these estimates are superior to other ones in the sense that they yield a smaller value of c c. At first sight the constant-free estimates follow a completely different pattern than the other indicators listed before since their derivation usually is based on a dual variational principle which originally is due to Prager and Synge [16]. Its simplest form can be stated as [13] u U inf ϱ sup (ϱ U, v). v Here, and (, ) denote the L 2 -norm and scalar product, respectively, u and U are two H 1 -functions which have the same boundary values and which correspond to the weak solution of a differential equation and its approximation, respectively, ϱ is any L 2 -vector field which has the same divergence as u in L 2, and v is any H 1 0-function. In this note we look at the constant-free error estimates from a different perspective. We show that they can be interpreted as an alternative way of expressing the residual and thus fit into the same framework as the other indicators listed before. Our approach is based on the following observations:

3 CONSTANT-FREE A POSTERIORI ERROR ESTIMATES 3 Given any approximate solution U of a differential equation, its residual R is an element of H 1 or a similar dual space. Still, if U is a finite element function, the residual admits an L 2 -representation with respect to the partition and its skeleton (cf. equation (2.1)). This gives some extra regularity. The L 2 -representation of the residual allows to lift it locally to some H(div)-space. This lifting requires some compatibility of the local fluxes, i.e. the representation of the residual on the elements and the skeleton. This compatibility can be ensured by either requiring Galerkin orthogonality or by extracting a suitable discrete residual. Collecting the local liftings leads to a global lifting of the residual to some broken H(div)-space. If the L 2 -representation of the residual consists of piece-wise polynomials, the lifting can be expressed in terms of functions from suitable broken Raviart-Thomas or Brezzi-Douglas-Marini spaces [5, Sections III.3.1, III3.2]. Moreover, this representation can easily be computed by sweeping through the elements (cf. Section 3). These observations yield a representation of the residual which can shortly be expressed in the form R, v = (ϱ T, v) + R, I T v (cf. equation (2.2)). Here I T is a quasi-interpolation operator with values in a suitable finite element space. To better understand this result, assume for simplicity that we have Galerkin orthogonality, i.e. R, I T v = 0, and distinguish two cases: The error is measured in the standard H 1 -norm or semi-norm and the residual is measured in the corresponding dual norm which is the standard H 1 -norm. The error is measured in a norm of the form {ε } 1 2 with 0 < ε 1 and the residual is measured in the corresponding dual norm. The first case corresponds to problems with dominant diffusion, the second case applies to problems with dominant reaction or dominant convection. In the first case, we conclude that ϱ T is a reliable error indicator with c = 1. A scaling argument shows that it can be bounded from above by the standard residual error indicator and therefore is efficient with a constant c which is a multiple of the corresponding constant of the residual indicator (cf. Proposition 2.2). In particular it remains an open question whether the condition of the constant-free error indicator is better than the one of the standard residual indicator or not.

4 4 R. VERFÜRTH In the second case, we conclude that ε 1 2 ϱ T is a reliable error indicator with c = 1. The scaling argument again shows that it can be bounded from above by a multiple of the robust residual error indicator introduced in [19, 20] (cf. Proposition 2.2). But, now the multiplying constant is no longer mesh-independent. Instead it is proportional to the mesh-peclét-number. Thus this simple form of the constantfree error indicator is not fully robust and thus inferior to the robust residual indicator of [19, 20]. This theoretical result is supported by the numerical results of Section 6 and [8]. The missing robustness is not a technical but a structural defect. It is due to the fact that the right-hand side of equation (2.2) only gives control on the gradient of the test-function v and not on the function itself. It can be remedied by taking sup v (ϱ T, v) {ε v 2 + v 2 } 1 2 as error indicator where the supremum is taken with respect to all H0-1 functions. This gives a constant-free robust error indicator. But, we do not know of any practical and efficient way to compute this quantity. Another possible remedy is to use { min ε 1 2 ϱt, [ ] } β div ϱ T + β ϱ T n as an error indicator. Here, the sum refers to the elements of the current partition and β and β are suitable weights similar to those used in [19, 20]. A variant of this approach is used in [6, 8, 11]. It yields an error indicator with c = 1, but it partially leaves the concept of constant-free indicators since the second term in the minimum may contain hidden constants which have to be estimated beforehand. The rest of this note is organized as follows. In the next section, we collect the required notations and state the main results. In Section 3, we construct the lifting of the residual and thus prove our first main result, Proposition 2.1. To simplify the exposition this is done in terms of piece-wise H(div)-functions. For practical computations these can be replaced in a well-defined way by piece-wise Raviart-Thomas or Brezzi-Douglas-Marini functions [5, Sections III3.1, III.3.2]. In Section 4, we discuss the reliability and efficiency of the error indicator and thus prove our second main result, Proposition 2.2. The upper bound of Proposition 2.2 contains a term I T R which measures the effect of violating the Galerkin orthogonality. In Section 5, we briefly indicate how to bound this term when it originates from SUPG-discretizations or nested iterative solvers. In Section 6, finally, we give a numerical example which supports the missing robustness of the estimator ε 1 2 ϱ T in the presence of dominant convection or reaction terms.

5 CONSTANT-FREE A POSTERIORI ERROR ESTIMATES 5 2. Notations and main results In what follows Ω denotes an open, bounded, connected set in R d with d 2 and polyhedral Lipschitz boundary Γ. The boundary splits into two disjoint parts Γ D and Γ N which are referred to as Dirichlet and Neumann boundary, respectively. The Neumann boundary may be empty, the Dirichlet boundary is assumed to be relatively closed and to have a positive (d 1)-dimensional Lebesgue measure. For any measurable set ω, we denote by L 2 (ω) and ω the standard Lebesgue space of square integrable functions and its standard norm, respectively. If ω = Ω, we omit the subscript Ω. The L 2 -norm of vector fields is defined as the L 2 -norm of the Euclidean norm of the vector field. Similarly, the L 2 -norm of a matrix-valued function is the L 2 -norm of the Frobenius norm of the matrix. The d- and (d 1)-dimensional Lebesgue measures are denoted by µ d and µ d 1, respectively. HD 1 (Ω) is the standard Sobolev space of L2 -functions having their derivatives in L 2 (Ω) and vanishing on the Dirichlet boundary Γ D. It is equipped with the norm v = { ε v 2 + v 2} 1 2 with 0 < ε 1. We are particularly interested in the two cases ε = 1 corresponding to diffusion dominated problems and ε 1 corresponding to problems with dominant reaction or convection. H 1 (Ω) denotes the dual space of HD 1 (Ω) and is equipped with the norm ϕ = ϕ, v sup v HD 1 (Ω)\{0} v where, stands for the standard duality pairing. We denote by T a partition of the domain Ω into polyhedral subdomains which satisfies the following conditions: The closure Ω of Ω is the union of all elements in T. The Dirichlet boundary Γ D is the union of (d 1)-dimensional faces of elements in T. Any two elements in T are either disjoint or share a vertex or a complete k-dimensional face with 1 k d 1 (admissibility). Every element in T is either a simplex or a parallelepiped (affine equivalence). These conditions could be weakened, but their present form simplifies the exposition. In addition, some estimates will depend on the shape parameter C T of T which, as usual, is defined as the maximal ratio of the diameter of any element in T over the diameter of the largest ball inscribed into that element. We denote by N and E the collection of all vertices and all (d 1)- dimensional faces (faces in short) corresponding to T. A subscript, E, Ω, Γ D, or Γ N to any of these sets indicates that only members in the

6 6 R. VERFÜRTH respective set are considered, i.e., N and E, e.g., denote the vertices and faces, resp. of a given element. The set Σ = E E E is the skeleton of the partition T. With every vertex z N we associate the sets ω z and σ z which are the union of all elements and all faces, resp. that have z as a vertex. For every element and every vertex z we denote by h and h z the diameters with respect to the Euclidean norm of the sets and ω z, respectively. With every vertex z N, we associate its nodal shape function λ z which is globally continuous, element-wise a polynomial of degree 1 and which satisfies λ z (z) = 1 and λ z (x) = 0 for all x N \ {z}. Notice that ω z is the support of λ z and that λ Σ z = σ z λ z. We define a quasi-interpolation operator I T on L 2 (Ω) by I T v = ω z λ z v if z N Ω N ΓN, λ z v z with v z = ω z λ z z N 0 if z N ΓD. It maps HD 1 (Ω) into the space of continuous piece-wise linear functions which vanish on the Dirichlet boundary. For any connected bounded open set ω R d with Lipschitz boundary γ, the space H(div; ω) consists of all vector fields σ L 2 (ω) d with div σ L 2 (ω). The space of all vector fields in L 2 (Ω) d which are element-wise in H(div; ) is denoted by H T (div). With these notations our first main result can be stated as follows: Proposition 2.1. Assume that the residual R H 1 (Ω) admits an L 2 -representation, i.e., there are functions r L 2 (Ω) and j L 2 (Σ) such that (2.1) R, v = rv + jv holds for all v HD 1 (Ω). Then there is a vector field ϱ T such that (2.2) R, v = ϱ T v + R, I T v Ω Ω Σ H T (div) holds for all v HD 1 (Ω). The vector field ϱ T can explicitly be computed by sweeping through the elements of T. Moreover, ϱ T can be chosen from the Raviart-Thomas space RT k+1 or the Brezzi-Douglas-Marini space BDM k+2 if the functions r and j are piece-wise polynomials of degree k. Proposition 2.1 will be proved in the next section. Based on Proposition 2.1 we will prove in Section 4 our second main result:

7 CONSTANT-FREE A POSTERIORI ERROR ESTIMATES 7 Proposition 2.2. Under the hypotheses of Proposition 2.1 the dual norm of the residual can be bounded from above by (2.3) R ε 1 2 ϱt + I T R. Here, ϱ T is as in Proposition 2.1 and IT denotes the dual of the operator I T. The vector field ϱ T also satisfies the stability estimate { [ ] } (2.4) ε ϱt c 2 T ε 1 h 2 z r 2 ω z + c 2 Eε 1 h z j 2 2 σ z. z N The constant c E depends on the shape parameter C T. If the vector field ϱ T is chosen from a Raviart-Thomas or Brezzi-Douglas-Marini space, both constants depend on the shape parameter and on the polynomial degree of ϱ T. For a better understanding of Propositions 2.1 and 2.2 consider a conforming finite element discretization of the convection-diffusion equation ε u + a u + bu = f in Ω (2.5) u = 0 εn u = g on Γ D on Γ N as a model problem. Here, conforming means that the finite element space is a subspace of HD 1 (Ω) and contains the lowest order conforming space spanned by the nodal shape functions λ z, z N Ω N ΓD. Denoting by u T the computed approximate solution of the discrete problem, the residual R is given by { R, v = fv + gv ε ut v + a u T v + bu T v }. Ω Γ N Ω Integration by parts element-wise yields the representation (2.1) with r = f +ε u T a u T bu T for all elements T, j E = g εn u T for all Neumann faces E E ΓN, and j E = [εn E u T ] E for all interior faces E E Ω. Here, as usual, [ ] E denotes the jump across E in the direction of a unit vector n E orthogonal to E. If the discrete problem does not contain any stabilization terms of SUPG-type and if it is solved exactly, we have Galerkin orthogonality, i.e. R, I T v = 0 for all v H 1 D (Ω) and I T R = 0. Otherwise, the quantity I T R must be bounded separately. An in-depth study of this problem is beyond the scope of this article. In Section 5, however, we briefly indicate how to accomplish this task for SUPG-discretizations and nested iterative solvers. 3. H T (div)-lifting of the residual In this section we construct the vector field ϱ T and thus prove Proposition 2.1. We start with an auxiliary result.

8 8 R. VERFÜRTH Lemma 3.1. For every element and every pair of functions f L 2 () and g L 2 ( ) which satisfy the compatibility condition (3.1) f + g = 0, there is a vector field τ H(div; ) with (3.2) div τ = f in, τ n = g on. Here, n denotes the unit exterior normal of. The vector field τ can be chosen such that the stability estimate (3.3) τ c,1 h f + c,2 h 1 2 g holds with c,1 = 1 π, c,2 = 2π + 1 ( h µ d 1 ( ) ) 1 2. π µ d () If the functions f and g are piece-wise polynomials of degree k, the vector field τ can be chosen from the Raviart-Thomas space RT k or the Brezzi-Douglas-Marini space BDM k+1 such that properties (3.2) and (3.3) remain valid. In this case both constants depend on the polynomial degree k and the shape parameter C T. Proof. The existence of τ is well-known and follows from [5, Lemmas III.1.1 and III.1.2] and [10, Lemma 2.1] or [2]. The stability result (3.3) should also be well-known, but we could not find a reference. Therefore we prove it in what follows. We first consider the analytical case. Due to the compatibility condition (3.1) there is a unique function v H 2 () which solves the Neumann problem v = f in, n v = g on, v = 0. For every w H 1 () it satisfies (3.4) v w = fw + The vector field τ = v obviously fulfils conditions (3.2). In order to prove the stability result we insert w = v in equation (3.4). This yields τ 2 = fv + gv f v + g v. Since v has vanishing mean-value on, we know from the Poincaré inequality [1, 15] that gw. v 1 π h v = 1 π h τ.

9 CONSTANT-FREE A POSTERIORI ERROR ESTIMATES 9 Together with the trace inequality [17, Corollary 4.5 and Remark 4.6] this gives for every face E contained in v 2 E µ d 1(E) µ d () v 2 + 2h µ d 1 (E) v v µ d () 2π + 1 h µ d 1 (E) h π 2 τ 2 µ d (). Combining these estimates proves the stability result in this case. Next we consider the discrete case where f and g are piece-wise polynomials of degree k. We now set τ = J v where J is the interpolation operator of [5] which maps H 1 () d into the corresponding space RT k or BDM k+1. (Note that J is labelled ρ in [5].) Its properties imply that τ fulfils conditions (3.2). Since J uses moments on the boundary of, we cannot expect to bound τ in terms of v. Still, from [5, Proposition III.3.6] we know that τ v + v τ v + ch 2 v with a constant c which depends on the polynomial degree k and the shape parameter C T. Hence, it remains to prove that the second term on the right-hand side of this estimate can be bounded by the right-hand side of (3.2) with appropriate constants depending on k and C T. We achieve this with the help of a scaling argument. Denote by F : x a + B x an orientation preserving affine transformation of the reference simplex or reference cube onto and set f = µ d() µ d ( ) b f F on and ĝ = µ d 1(F ( E)) b µ d 1 ( E) b g F ( E) b F on every face Ê of. The compatibility condition (3.1) implies that b f + b ĝ = 0. Hence, there is a unique function v H 2 ( ) which solves the Neumann problem v = f on, n v = ĝ on, v = 0, where n is the unit exterior normal of. The properties of the Piola transformation τ µ d( ) b B µ d () τ imply that v = µ d( ) b B µ d () v and [5, Lemma III1.7] ( 2 µd ( v ) ) 1 2 B 2 B 1 µ d () 2 2 v b, where 2 denotes the spectral norm of matrices. Since is convex and since ĝ is a piece-wise polynomial, we have 2 v b ĉ 1 { f b + ĝ H 1 2 ( b ) } ĉ1 { f b + ĉ 2 ĝ b } ĉ 1 {( µd () µ d ( ) ) 1 2 f + ĉ 2 max be E b ( µd 1 (F (Ê) b µ d 1 (Ê) ) 1 2 g }

10 10 R. VERFÜRTH with a constant ĉ 2 which depends on the polynomial degree k. We conclude the proof by combining all these estimates and taking into account that B 2 B 1 2 only depends on the shape parameter C T. Remark 3.2. We stress that in the case of polynomial data f and g, problem (3.2) can directly be solved on the discrete level by considering the corresponding Galerkin formulation in the Raviart-Thomas or Brezzi-Douglas-Marini spaces. The Neumann problems and their solutions v and v are needed to prove the stability result (3.3) but not to compute the discrete version of τ. We now construct ϱ T piece-wise on the patches ω z by generalizing the idea of [3, Algorithm 9.3] and [4, 7]. Consider an arbitrary vertex z N and set R, λ z ω = z λ z r + σ z λ z j if z N Ω N ΓN, R z = ω z λ z ω z λ z 0 if z N ΓD. Since Ω has a Lipschitz boundary, the same applies to ω z. Hence, any pair of elements in ω z can be connected by a path of adjacent elements in ω z. Therefore, we can enumerate the elements in ω z from 1 to n z = { ω z } such that (see Figure 3.1): For every i {1,..., n z 1} the elements i and i+1 share a face E i which is contained in σ z. Either nz and 1 share a face E nz which is contained in σ z and which is different from the faces E 1,..., E nz 1, or nz has a face E nz which is contained in σ z Γ. If z is on the Dirichlet boundary, the face E nz is also contained in the Dirichlet boundary Figure 3.1. Enumeration of the elements in ω z for a vertex z in Ω (left) and on Γ (right); the face E nz is indicated by a bold line. Set α 1 = 1 { } λ z (r R z ) + λ z j µ d 1 (E 1 ) 1 1 σ z

11 CONSTANT-FREE A POSTERIORI ERROR ESTIMATES 11 and, for 2 i n z, 1 { } α i = λ z (r R z ) + λ z j + µ d 1 (E i 1 )α i 1. µ d 1 (E i ) i ( i σ z)\e i 1 Then we have µ d 1 (E nz )α nz = ω z λ z (r R z ) + σ z λ z j. If z is not on the Dirichlet boundary, the definition of R z implies α nz = 0. If, on the other hand, z is on the Dirichlet boundary, α nz in general is not zero. Due to the definition of α 1,..., α nz, we may apply Lemma 3.1 to the sets 1,..., nz with λ z j on ( 1 σ z ) \ E 1, f = λ z (r R z ) in 1, g = λ z j α 1 on E 1, 0 on 1 \ σ z and, for i {2,..., n z }, λ z j on ( i σ z ) \ (E i 1 E i ), α i 1 on E i 1, f = λ z (r R z ) in i, g = λ z j α i on E i, 0 on i \ σ z and thus obtain vector fields ϱ i H(div; i ) for 1 i n z. We extend every ϱ i by zero outside i and set ϱ z = n z i=1 ϱ i. Consider an arbitrary function v in HD 1 (Ω). Integration by parts on 1 yields ϱ 1 v = ϱ 1 (v v z ) 1 1 = div ϱ 1 (v v z ) + n 1 ϱ 1 (v v z ) 1 1 = λ z (r R z )(v v z ) + λ z j(v v z ) 1 1 σ z α 1 (v v z ). E 1 Similarly we obtain for every i {2,..., n z } ϱ i v = ϱ i (v v z ) i i = div ϱ i (v v z ) + n i ϱ i (v v z ) i i = λ z (r R z )(v v z ) + λ z j(v v z ) i ( i σ z)\e i 1 + α i 1 (v v z ) α i (v v z ). E i 1 E i

12 12 R. VERFÜRTH Summation with respect to i yields ϱ z v = λ z (r R z )(v v z ) + λ z j(v v z ) ω z ω z σ z α nz (v v z ). E nz If z is not on the Dirichlet boundary, we have α nz = 0. If, on the other hand, z is on the Dirichlet boundary, E nz is contained in Γ D and both v and v z vanish on E nz. Hence, the contribution of E nz vanishes in both cases and we obtain ϱ z v = ω z λ z (r R z )(v v z ) + ω z λ z j(v v z ). σ z Still, the definition of v z implies that λ z R z (v v z ) = 0. ω z Hence, we arrive at ϱ z v = ω z λ z r(v v z ) + ω z λ z j(v v z ) σ z for every vertex z N. Now, we extend every ϱ z by zero outside ω z and set ϱ T = z N ϱ z. Taking into account that the functions λ z form a partition of unity, i.e. z N λ z = 1, summation with respect to all vertices yields ϱ T v = ϱ z v Ω z N Ω = ϱ z v z N ω z = { } λ z r(v v z ) + λ z j(v v z ) z N ω z σ z = { } λ z r(v v z ) + λ z j(v v z ) z N Ω Σ = R, v I T v for all v HD 1 (Ω). This proves equation (2.2) and Proposition Reliability and efficiency In this section we prove Proposition 2.2. The reliability, inequality (2.3), is an obvious consequence of the representation (2.2) and the definition of the dual norm. For the proof of the efficiency, inequality (2.4), we apply Lemma 3.1 and the stability estimate (3.3) at the different stages of the construction of ϱ T.

13 CONSTANT-FREE A POSTERIORI ERROR ESTIMATES 13 Consider an arbitrary vertex z N. Using the notations of Section 3, inequality (3.3) implies for the vector field ϱ 1 and the element 1 ϱ c 2 1,1h 2 1 λ z (r R z ) c 2 1,2h 1 λ z j 2 1 σ z + 4c 2 1,2h 1 µ d 1 (E 1 ) α 1 2. The definition of α 1 and the Cauchy-Schwarz inequality, on the other hand, yield h 1 µ d 1 (E 1 ) α 1 2 µ d ( 1 ) 2 h 2 µ d 1 (E 1 )h 1 λ z (r R z ) Similarly, we obtain for all i {2,..., n z } + 2 µ d 1( 1 σ z ) h 1 λ z j 2 µ d 1 (E 1 ) 1 σ z. and ϱ i 2 i 2c 2 i,1h 2 i λ z (r R z ) 2 i + 4c 2 i,2h i λ z j 2 ( 1 σ z)\e i 1 + 4c 2 i,2h i µ d 1 (E i 1 ) α i c 2 i,2h i µ d 1 (E i ) α i 2 h i µ d 1 (E i ) α i 2 µ d ( i ) 3 h 2 µ d 1 (E i )h i λ z (r R z ) 2 i i Adding all estimates we arrive at + 3 µ d 1(( i σ z ) \ E i 1 ) h i λ z j 2 ( µ d 1 (E i ) i σ z)\e i h i µ d 1 (E i 1 ) h i 1 µ d 1 (E i ) h i 1 µ d 1 (E i ) α i 1 2. ϱ z 2 ω z c 2 z,1h 2 z λ z (r R z ) 2 ω z + c 2 z,2h z λ z j 2 σ z, where the constants c z,1 and c z,2 depend on n z and the quantities max 1 i n z c i,1, max 1 i n z c i,2, µ d ( i ) µ d 1 ( i σ z ) max, max, 1 i n z µ d 1 (E i )h i 1 i n z µ d 1 (E i ) h i µ d 1 (E j ) max 1 i<j n z h j µ d 1 (E i ). From the definition of R z and the Cauchy-Schwarz inequality we conclude that ( ) λ z R z 2 ω ω z = z λ 2 z ) 2 λ z r + λ z j 2 (ω z λ z ω z σ z 2 µ d(ω z ) ( ω z λ z ) λ z r 2 ω z + 2 h zµ d 1(σ z ) (ω z λ z ) h 1 z λ z j 2 σ z.

14 14 R. VERFÜRTH Since every vector field ϱ z vanishes outside ω z and since every element intersects N patches ω z, we have ϱ T 2 = T ϱ T 2 = ϱ z 2 T z N ( max N ) ϱ z 2 T T z N = ( max N ) ϱ z 2 ω T z. Combining these estimates proves the stability result (2.4) and Proposition 2.2. z N 5. Estimation of I T R In this section we briefly indicate how to bound the term I T R when it originates from SUPG-discretizations and nested iterative solvers. We first consider the case of an exact solution of an SUPG-discretization. To simplify the exposition, we assume that the data in problem (2.5) are all piece-wise polynomials and that consequently the function ϱ T belongs to a suitable Raviart-Thomas or Brezzi-Douglas-Marini space. If this condition is violated additional data errors must be added to the upper bound of I T R. From [9, 12, 20] we conclude that R, I T v = δ ra (I T v), T where δ 0 are the stabilization parameters. To bound this term, consider an arbitrary function v HD 1 (Ω) and define the function w L 2 (Ω) by w = δ a (I T v) for all T. The property z λ z = 1, the definitions of ϱ T, R z and I T and a weighted Cauchy-Schwarz inequality imply that R, I T v = λ z (r R z )w + λ z R z w T z N T z N = ( div ϱ T )w + R, I T w Ω { ε 1 h 2 div ϱ T 2 T + R I T w. } 1 2 { T } 1 εh 2 w 2 2

15 CONSTANT-FREE A POSTERIORI ERROR ESTIMATES 15 Standard inverse estimates yield for every element div ϱ T c I h 1 ϱ T, w δ a L () min{ε 1 2, ci h 1 } I T v, I T w {ε c 2 Ih 2 + 1} 1 2 IT w, where c I and c I both depend on the shape parameter C T and where c I in addition depends on the polynomial degree of ϱ T. The definition of I T implies that I T w λ z w z w ωz w ωz z N ω z λ z z N z N λ z 2 holds for every element. Moreover there is a constant ĉ I which only depends on the shape parameter C T such that the inequality { ε c 2 I h } min{ε 1 h 2, c 2 I} 2 c 2 min{ε 1 h 2, c2 I } I min{ε 1 h 2, c 2 I } ĉ2 I holds for every pair of elements, which share a vertex. Finally, every element intersects at most N patches ω z. Combining all I these estimates and denoting by I T L = sup T v v the operator norm v of I T, we obtain the upper bound I T R ε 1 2 ϱt max T (h 1 δ )c I a L (Ω) I T L + R max T (δ h 1 N )ĉ I a L (Ω) I T L. Since, according to [9, 12], the stability parameters δ should be less than the local mesh-size h, we may assume that max (δ h 1 ) 1 T 2 max{ c I a L (Ω) I T L, max ( N } 1. )ĉ I a L T (Ω) I T L The above estimate of I T R and Proposition 2.2 then imply that R 3ε 1 2 ϱt. Thus, in the case of an SUPG-discretization, we have to pay for the missing Galerkin orthogonality by a factor 3 in the upper bound for the residual. We now consider the case of a nested iterative solver as it is often used within an adaptive finite element code (cf. e.g. [3, V.4]). Such an algorithm is associated with a sequence T 0,..., T l of partitions obtained by repeated adaptive refinement with T 0 and T l being the coarsest and finest partition, respectively. It typically has the following properties: The discrete problem corresponding to T 0 is solved exactly.

16 16 R. VERFÜRTH The discrete problem corresponding to T k, k 1, is solved approximately with an iterative solver. The computed approximate solution corresponding to T k 1 is used as starting value and the iteration is stopped once the initial residual is reduced by a factor θ k. To simplify the exposition, we assume that the exact solution of the discrete problem corresponding to T 0 satisfies the Galerkin orthogonality. Moreover, for every k {0,..., l}, we denote by X k the finite element space corresponding to T k, by R k the residual of the computed approximate solution of the discrete problem corresponding to X k and by ϱ k the corresponding vector field according to Proposition 2.1. The above assumptions and Proposition 2.2 then imply that I T 0 R 0 = 0 and for k 1 I T R k, v k k R k I Tk L sup v k X k v k R k 1, v k θ k I Tk L sup v k X k v k θ k I Tk L R k 1 { θ k I Tk L ε 1 2 ϱk 1 + I T k 1 R k 1 }. By induction this estimate and Proposition 2.2 prove that l 1 l R l ε 1 2 ϱl + ε 1 2 ϱk θ m I Tm L. k=0 m=k+1 Thus, in the case of a nested iterative solver, the residual R l can be controlled by the vector fields ϱ 0,..., ϱ l and the stopping tolerances θ 1,..., θ l. 6. A numerical example We consider linear finite element discretizations of problem (2.5) with the following data: The domain Ω is the unit square (0, 1) 2 with empty Neumann boundary Γ N, the convection a vanishes, the reaction is b = 8 x 2 tanh(ε 1 2 ( x ))2 and the load f is chosen such that the exact solution of (2.5) is u = ε 1 2 [tanh(ε 1 2 ( x 2 1)) 4 tanh(ε 1 2 3)]. 4 The partitions T are Courant triangulations and consist of right-angled isosceles triangles with short sides of length h = 2 l with 3 l 6. The discrete problems are solved with a nested multigrid algorithm with Gauss-Seidel smoothing and stopping tolerance θ k = 0.01 on all levels k. Table 6.1 gives the effectivity indices 1 ε 2 ϱ T u u T for various combinations of diffusion parameter ε and mesh-size h. It supports the theo- retical result that the estimator ε 1 2 ϱ T is not fully robust.

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18 18 R. VERFÜRTH [16] W. Prager and J. L. Synge, Approximations in elasticity based on the concept of function space, Quart. Appl. Math. 5 (1947), [17] A. Veeser and R. Verfürth, Explicit upper bounds for dual norms of residuals, report, Ruhr-Universität, Bochum, October [18] R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh- Refinement Techniques, Teubner-Wiley, Stuttgart, [19], Robust a posteriori error estimators for a singularly perturbed reaction-diffusion equation, Numer. Math. 78 (1998), no. 3, [20], Robust a posteriori error estimates for stationary convection-diffusion equations, SIAM J. Numer. Anal. 43 (2005), no. 4, Fakultät für Mathematik, Ruhr-Universität Bochum, D Bochum, Germany address: ruediger.verfuerth@ruhr-uni-bochum.de