Chapter 6 A posteriori error estimates for finite element approximations 6.1 Introduction

Chapter 6 A posteriori error estimates for finite element approximations 6.1 Introduction The a posteriori error estimation of finite element approximations of elliptic boundary value problems has reached some state of maturity, as it is documented by a variety of monographs on this subject (cf., e.g., [1, 2, 3, 4, 5]). There are different concepts such as residual type a posteriori error estimators, hierarchical type a posteriori error estimators, error estimators based on local averaging, error estimators based on the goal oriented weighted dual approach (cf., in particular, [3]). In this chapter, we will focus on residual type a posteriori error estimators and follow the exposition in [5]. We shall deal with the following model problem: Let Ω be a bounded simply-connected polygonal domain in uclidean space lr 2 with boundary Γ = Γ D Γ N, Γ D Γ N = and consider the elliptic boundary value problem (6.1) Lu := div (a grad u) = f in Ω, u = 0 on Γ D, n a grad u = g on Γ N, where f L 2 (Ω), g H 1/2 (Γ N ) and a = (a ij ) 2 i,j=1 is supposed to be a matrix-valued function with entries a ij L (Ω), that is symmetric, i.e., a ij (x) = a ji (x) f.a.a. x Ω, 1 i, j 2, and uniformly positive definite in the sense that for almost all x Ω 2 a ij (x) ξ i ξ j α ξ 2, ξ R 2, α > 0. i,j=1 The vector n denotes the exterior unit normal vector on Γ N. We further set ᾱ := max 1 i,j 2 a ij. Setting H 1 0,Γ D (Ω) := { v H 1 (Ω) v ΓD = 0 }, the weak formulation of (6.1) is as follows: Find u H 1 0,Γ D (Ω) such that (6.2) a(u, v) = l(v), v H0,Γ 1 D (Ω), 1

2 Ronald H.W. Hoppe where a(v, w) := a grad v grad w dx, v, w H 1 0,Γ D (Ω), l(v) := Ω Ω f v dx + Γ N g v dσ, v H 1 0,Γ D (Ω). Given a geometrically conforming simplicial triangulation T h of Ω, we denote by S 1,ΓD (Ω; T h ) := { v h H 1 0,Γ D (Ω) v h P 1 (), T h } the trial space of continuous, piecewise linear finite elements with respect to T h. Note that P k (), k 0, denotes the linear space of polynomials of degree k on. In the sequel we will refer to N h (D) and h (D), D Ω as the sets of vertices and edges of T h on D. The conforming P1 approximation of (6.1) reads as follows: Find u h S 1,ΓD (Ω; T h ) such that (6.3) a(u h, v h ) = l(v h ), v h S 1,ΓD (Ω; T h ). Now, assuming that ũ h S 1,ΓD (Ω; T h ) is some iterative approximation of u h S 1,ΓD (Ω; T h ), we are interested in the total error e := u ũ h, which is the sum of the discretization error e d := u u h and the iteration error e it := u h ũ h. It is easy to see that the total error e is in V := H 1 0,Γ D (Ω) and satisfies the error equation (6.4) a(e, v) = r(v), v V, where r( ) stands for the residual with respect to the computed approximation ũ h (6.5) r(v) := f v dx + g vdσ a(ũ h, v), v V. Ω Γ N We are interested in a cheaply computable a posteriori error estimator η of the total error e consisting of elementwise error contributions η, T h and edgewise error contributions η, h, in the sense that (6.6) η 2 = T h η 2 + h η 2, which does provide a lower and an upper bound for e according to (6.7) γ η e 1,Ω Γ η

Finite lement Methods 3 with constants 0 < γ Γ depending only on the ellipticity constants and on the shape regularity of the triangulation T h. We may use the local error terms η and η as a criterion for local refinements of the elements T h. Among several refinement strategies, the so-called mean-value strategy is as follows: Compute the mean values η := 1 (6.8) η, n T h η := 1 n h η, where n := card T h and n := card h. Mark an element T h and an edge h for refinement, if (6.9) η σ η, η σ η, where 0 < σ 1 is some appropriate safety factor, e.g., σ = 0.9. Definition 6.1 fficient and reliable a posteriori error estimators An a posteriori error estimator η satisfying (6.10) e 1,Ω Γ η is called reliable, since it ensures a sufficient refinement in the sense that the H 1 -norm of the total error e will be bounded by a quantity of the same order of magnitude as a user-prescribed accuracy, if this accuracy is tested by η. On the other hand, an a posteriori error estimator η for which (6.11) γ η e 1,Ω is said to be efficient, since it underestimates the H 1 -norm of the total error e and thus prevents too much refinement. 6.2 Residual based a posteriori error estimators The residual based a posteriori error estimator can be derived by viewing the residual as an element of the dual space V and evaluating it with respect to the dual norm. 6.2.1 Upper bound for the total error An important tool in the construction of an upper bound for the total error is Clément s interpolation operator which is defined as follows:

4 Ronald H.W. Hoppe Definition 6.2 Clément s interpolation operator For p N h (Ω) N h (Γ N ) we denote by ϕ p the basis function in S 1,ΓD (Ω; T h ) with supporting point p and we refer to D p as the set (6.12) D p := { T h p N h (T ) }.0 D p p Fig. 6.1: Clément s interpolation operator (definition) We refer to π p as the L 2 -projection onto P 1 (D p ), i.e., (π p (v), w) 0,Dp = (v, w) 0,Dp, w P 1 (D p ), where (, ) 0,Dp stands for the L 2 -inner product on L 2 (D p ) L 2 (D p ). Then, Clément s interpolation operator P C is defined as follows (6.13) P C : L 2 (Ω) S 1,ΓD (Ω, T h ), P C v := π P (v) ϕ P. p N h (Ω) N h (Γ N ) In order to establish local approximation properties of Clément s interpolation operator, for T h and h (Ω) h (Γ N ) we introduce the sets (6.14) (6.15) D (1) := { T h N h ( ) N h () }, D (1) := { T h N h () N h ( ) }.

Finite lement Methods 5 Fig. 6.2: Clément s interpolation operator (properties) Using the affine equivalence of the elements and the Bramble- Hilbert Lemma one can show: Theorem 6.1 Approximation properties of Clément s interpolation operator For T h and h (Ω) h (Γ N ) let D (1) and D(1) be given by (6.14) and let P C be Clément s interpolation operator as given by (6.13). Then, there exist constants C ν > 0, 1 ν 5, depending only on the shape regularity of T h such that for all v S 1,ΓD (Ω; T h ): (6.16) (6.17) (6.18) (6.19) (6.20) P C v 0, C 1 v (1) 0,D, P C v 0, C 2 v (1) 0,D, grad P C v 0, C 3 grad v (1) 0,D v P C v 0, C 4 h v (1) 1,D, v P C v 0, C 5 h 1/2 v 1,D (1),, where h := diam and h :=. Further, there exist constants C 6, C 7 > 0 depending only on the shape regularity of T h such that (6.21) (6.22)( h (Ω) h (Γ N ) ( v 2 ) 1/2 C µ,d (1) 6 v µ,ω, 0 µ 1, T h v 2 ) 1/2 C µ,d (1) 7 v µ,ω, 0 µ 1. Proof. We refer to [5]. We have now provided all prerequisites to establish an upper bound for the total error e measured in the H 1 -norm. For functions v h W 0 (Ω; T h ) we further refer to [v h ] J as the jump across the common

6 Ronald H.W. Hoppe edge h (Ω) of two adjacent elements 1, 2 T h [v h ] J := v h 1 v h 2. Theorem 6.2 Upper bound for the total error There exist constants Γ R, Γ osc > 0 and Γ it > 0 depending only on the elipticity constants and the shape regularity of T h such that (6.23) e 1,Ω Γ R η R + Γ osc osc + η it e it 1,Ω, where the element and edge residuals are given by η R := 3 ν=1 η (ν) R, η (1) R := ( T h h 2 T π h f Lũ h 2 0,) 1/2, η R := ( h (Γ N ) η (3) R := ( h (Ω) h π h g n a grad ũ h 2 0,) 1/2, h [n a grad ũ h ] J 2 0,) 1/2, and osc stands for the data oscillations osc := ( T h osc 2 + h (Γ N ) osc 2 ) 1/2, osc := h T f π h f 0,, osc := h g π h g 0,. Proof. Setting v = e in (6.4), we obtain (6.24) α e 2 1,Ω a(e, e) = r(e) = r(p C e) + r(e P C e). Taking advantage of (6.3), for the first term on the right-hand side of (6.24) we get r(p C e) = f P C e dx + g P C e dσ a(ũ h, P C e) = = Ω Γ N a (u h ũ h, P C e). T h

Finite lement Methods 7 Using (6.19), the Schwarz inequality, and observing (6.22), it follows that (6.25) r(p C e) α C 3 u h ũ h 1, e (1) 1,D T h α C 3 ( u h ũ h 2 1,) 1/2 ( e 2 ) 1/2 1,D (1) T h T h α C 3 C 6 e it 1,Ω e 1,Ω. On the other hand, for the second term on the right-hand side of (6.24), Green s formula yields r(e P C e) = f (e P C e) dx + g (e P C e) dσ + Ω Γ N + div a grad ũ h (e P }{{} C e) dx T h = Lũ h n a grad ũ h (e P C e) dσ = T h = T h (π h f Lũ h ) (e P C e) dx + + h (Ω) + h (Γ N ) [n a grad ũ h ] J (e P C e) dσ + (π h g n a grad ũ h ) (e P C e) dσ + + (f π h f) (e P C e) dx + T h + h (Γ N ) (g π h g) (e P C e) dσ. In view of (6.17),(6.18) and (6.22),(6.23), it follows that (6.26) r(e P C e) C 1 C 6 ( h 2 π h f Lũ h 2 0,) 1/2 e 1,Ω + T h + C 2 C 7 ( h [n a grad ũ h ] J 2 0,) 1/2 e 1,Ω + h (Ω)

8 Ronald H.W. Hoppe + C 2 C 7 ( h (Γ N ) h π h g n a grad ũ h 2 0,) 1/2 e 1,Ω + + C 1 C 6 ( T h h 2 f π h f 2 0,) 1/2 e 1,Ω + + C 2 C 7 ( h (Γ N ) h g π h g 2 0,) 1/2 e 1,Ω. Using (6.25),(6.26) in (6.24), the assertion follows with Γ R = Γ osc := α 1 max(c 1 C 6, C 2 C 7 ) and Γ it := α 1 α C 3 C 6. For the construction of a lower bound we will now show that the local contributions η (ν) R, := η(ν) R, T h, 1 ν 3 of the residual based error estimator η R do locally provide lower bounds for the total error e. For this purpose we need appropriate localized polynomial functions defined on the elements of the triangulation and the edges h (Ω) h (Γ N ), respectively. Such functions are given by the triangle-bubble functions ψ and the edge-bubble functions ψ. In particular, denoting by λ i, 1 i 3, the barycentric coordinates of T h, then the triangle-bubble function ψ is defined by means of (6.27) ψ := 27 λ 1 λ 2 λ 3. Note that supp ψ = int, i.e., ψ = 0, T h. On the other hand, for h (Ω) h (Γ N ) and T h such that and p i N h (T ), 1 i 2, we introduce the edge-bubble functions ψ according to (6.28) ψ := 4 λ 1 λ 2. Note that ψ = 0 for h (),. The bubble functions ψ and ψ have the following important properties that can be easily verified taking advantage of the affine equivalence of the elements:

Finite lement Methods 9 Lemma 6.1 Basic properties of the bubble functions. Part I There exist constants C ν > 0, 8 ν 12, depending only on the shape regularity of the triangulations T h such that (6.29) p h 2 0, C 8 p 2 h ψ dx, p h P 1 (), (6.30) p h 2 0, C 9 p 2 h ψ dσ, p h P 1 (), (6.31) (6.32) (6.33) p h ψ 1, C 10 h 1 p h 0,, p h P 1 (), p h ψ 0, C 11 p h 0,, p h P 1 (), p h ψ 0, C 12 p h 0,, p h P 1 (). For functions p h P 1 (), h (Ω) h (Γ N ) we further need an extension p h L2 () where T h such that. For this purpose we fix some,, and for x denote by x that point on such that (x x ). For p h P 1 () we then set (6.34) p h := p h (x ). Fig. 6.3: Level lines of the extension p h Further, for h (Ω) h (Γ N ) we define D as the union of elements T h containing as a common edge (6.35) D := { T h h () }.

10 Ronald H.W. Hoppe Fig. 6.4: The set D We have the following properties of the extensions: Lemma 6.2 Basic properties of the bubble functions. Part II There exist constants C ν > 0, 13 ν 14, depending only on the shape regularity of the triangulations T h such that (6.36) (6.37) p h ψ 1,D p h ψ 0,D C 13 h 1/2 p h 0,e, p h P 1 (), C 14 h 1/2 p h 0,, p h P 1 (). Further, there exists a constant C 15 > 0 independent of h, h such that for all v V and µ = 0, 1: (6.38) ( h 1 µ ) 1/2 C 15 ( h 1 µ v 2 µ,) 1/2. T h h (Ω) h (Γ N ) v 2 µ,d We are now able to prove that up to higher order terms the estimator η R also does provide a lower bound for the total error e: Theorem 6.3 Lower bound for the total error There exist constants γ R, γ > 0, depending only on ᾱ and on the shape regularity of T h such that (6.39) γ R η R γ osc e 1,Ω, where η R and osc are given as in the previous theorem. The theorem can be proved by a series of results which establish upper bounds for the local contributions η (ν) R,T, ν 3 of the estimator η R. Lemma 6.3 Upper bounds for the local contributions (i) Let T h. Then there holds: (6.40) h π h f L ũ h 0, ᾱ C 8 C 10 e 1, + C 8 C 11 h f π h f 0,.

(ii) Let h (Ω). Then there holds: Finite lement Methods 11 (6.41) h 1/2 [n a grad ũ h ] J 0, ᾱc 9 C 13 e 1,D + +C 9 C 14 h f π h f 0,D + C 9 C 14 h π h f Lũ h 0,D. (iii) Let h (Γ N ). Then there holds: (6.42) h 1/2 π h g n grad ũ h 0, ᾱc 9 C 13 e 1,D +C 9 C 14 h f π h f 0,D + C 9 C 12 h 1/2 g π h g 0, + + C 9 C 14 h π h f Lũ h 0,D. Proof of (i) For the proof of (6.40) we set p h := π h f. Observing ψ = 0, by Green s formula (6.43) a (ũ h, p h ψ ) = + n a grad ũ h p h ψ }{{} = 0 div (a grad ũ h ) p h ψ dx + dσ. Then, using (6.30),(6.32) and (6.33) and taking advantage of (6.4),(6.43), it follows that π h f L ũ h 2 0, C 8 (π h f L ũ h ) π h ψ dx = = C 8 ( f π h ψ dx a (ũ h, π h ψ ) + + ) (π h f f) π h ψ dx = = C 8 (a (e, π h ψ ) + ) (π h f f) π h ψ dx C 8 C 10 α h 1 e 1, p h 0, + C 8 C 11 π h f f 0, p h 0,, from which (6.40) can be easily deduced.

12 Ronald H.W. Hoppe Proof of (ii) We set p h := [n a grad ũ h ] J. Again, in view of ψ = 0,,, Green s formula gives (6.44) D n D a grad ũ h p h ψ dσ = = a D (ũ h, p h ψ ) + D div a grad ũ h }{{} = Lũ h p h ψ dx. If we use (6.31),(6.37) and (6.38) and observe (6.4) and (6.44), it follows that [n D C 9 = D a grad ũ h ] J 2 0, [n D [n D a grad ũ h ] J p h ψ dσ = a grad ũ h ] J p h ψ dσ = = C 9 (a D (ũ h, p h ψ ) + (f π h f) p h ψ dx + D f p h ψ dx + ) (π h f Lũ h ) p h ψ dx = D D = C 9 a D (e, p h ψ ) + ( + C 9 (f π h f) p h ψ dx + ) (π h f Lũ h ) p h ψ dx D D C 9 C 13 α h 1/2 e 1,D p h 0, + + C 9 C 14 h 1/2 f π hf 0,D p h 0, + + C 9 C 14 h 1/2 π h Lũ h 0,D p h 0,, from which we readily deduce (6.41).

Finite lement Methods 13 Proof of (iii) We set p h := π hg n a grad ũ h. Observing ψ = 0,, by Green s formula we obtain (6.45) n a grad ũ h p h ψ dσ = = D n D = a D (ũ h, p h ψ ) + a grad ũ h p h ψ dσ = D div a grad ũ h }{{} = Lũ h p h ψ dx. Now, using (6.31),(6.34),(6.37) and (6.38) and taking advantage of (6.4),(6.45), we get π h g n a grad ũ h 2 0, C 9 (π h g n a grad ũ h ) p h ψ dσ = = C 9 ( + D f p h ψ dx + (π h f f) p h ψ dx + g p h ψ dσ a D (ũ h, p h ψ ) + (π h g g) f p h ψ dσ D ) (π h f Lũ h ) p h ψ dx = D = C 9 a D (e, p h + ψ ) + C 9 ( (π h g g) f p h ψ dσ D (π h f f) p h ψ dx + ) (π h f Lũ h ) p h ψ dx D C 9 C 13 α h 1/2 e 1,D p h 0, + + C 9 C 14 h 1/2 π hf f 0,D p h 0, + + C 9 C 12 π h g g 0, p h 0, + + C 9 C 14 h 1/2 π hf Lũ h 0,D p h 0,,

14 Ronald H.W. Hoppe from which (6.41) follows easily. References [1] Ainsworth, M. and Oden, J.T. (2000). A Posteriori rror stimation in Finite lement Analysis. Wiley, Chichester. [2] Babuska, I. and Strouboulis, T. (2001). The Finite lement Method and its Reliability. Clarendon Press, Oxford. [3] Bangerth, W. and Rannacher, R. (2003). Adaptive Finite lement Methods for Differential quations. Lectures in Mathematics. TH-Zürich. Birkhäuser, Basel. [4] riksson,. and step, D. and Hansbo, P. and Johnson, C. (1995). Computational Differential quations. Cambridge University Press, Cambridge. [5] Verfürth, R. (1996). A Review of A Posteriori stimation and Adaptive Mesh- Refinement Techniques. Wiley-Teubner, New York, Stuttgart.