Basic Concepts of Adaptive Finite Element Methods for Elliptic Boundary Value Problems

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1 Basic Concepts of Adaptive Finite lement Methods for lliptic Boundary Value Problems Ronald H.W. Hoppe 1,2 1 Department of Mathematics, University of Houston 2 Institute of Mathematics, University of Augsburg Summer School Challenges in Applied Control and Optimal Control Basque Center for Applied Mathematics Derio, July 4-8, 2011

2 The Loop in Adaptive Finite lement Methods (AFM) Adaptive Finite lement Methods (AFM) consist of successive loops of the cycle SOLV: SOLV = STIMAT = MARK = RFIN Numerical solution of the F discretized problem STIMAT: Residual and hierarchical a posteriori error estimators rror estimators based on local averaging Goal oriented weighted dual approach Functional type a posteriori error bounds MARK: RFIN: Strategies based on the max. error or the averaged error Bulk criterion for AFMs Bisection or red/green refinement or combinations thereof

3 A Posteriori rror stimation I For a closed subspace V H 1 (Ω) we assume a(, ) : V V R to be a bounded, V-elliptic bilinear form, i.e., a(v,w) C v k,ω w k,ω, v,w V, a(v,v) γ v 2 k,ω, v V, for some constants C > 0 and γ > 0. We further assume l V where V denotes the algebraic and topological dual of V and consider the variational equation: Find u V such that a(u,v) = l(v), v V. It is well-known by the Lax-Milgram Lemma that under the above assumptions the variational problem admits a unique solution.

4 A Posteriori rror stimation II xample. The standard example is Poisson s equation: Let Γ = Γ D Γ N,Γ D Γ N = and V := {v H 1 (Ω) v ΓD = 0} and a(v,w) := ( v, w) 0,Ω, v,w V, l(v) := (f,v) 0,Ω + (g,v) 0,ΓN, v V, where f L 2 (Ω) and g L 2 (Γ N ). Then, the variational equation represents the weak form of Poisson s problem u = f in Ω, u = 0 on Γ D, n u = g on Γ N, where n stands for the unit outward normal on Γ N.

5 A Posteriori rror stimation III xample cont d. We remark that in case Γ N = we have V = H 1 0(Ω) (pure homogeneous Dirichlet boundary conditions), whereas in case Γ D = the appropriate function space is V = {v H 1 (Ω) (v,χ Ω ) 0,Ω = 0}, where χ Ω stands for the characteristic function of Ω. In these cases, the V-ellipticity of the bilinear form a(, ) follows from the Poincaré-Friedrichs inequalities ( ) v 0,Ω C v 1,Ω + vds, ( v 0,Ω C v 1,Ω + Γ ) vdx. Ω

6 A Posteriori rror stimation IV An important issue in the theory of partial differential equations is the regularity of a solution. For instance, considering Poisson s problem, it is well-known that for a convex domain Ω R 2 with piecewise smooth boundary Γ and f L 2 (Ω) the weak solution u V satisfies u V H 2 (Ω) and u 2,Ω C f 0,Ω. However, there is less regularity, if the domain Ω is no longer convex. A classical example is the so-called L-shaped domain: Consider the Poisson equation in Ω := ( 1, +1) (0,1) ( 1,0) ( 1,0], Γ D := {0} [ 1,0] [0,1] {0}, Γ N := Γ \ Γ D, and assume f 0 and g such that u(r,ϕ) = r 2/3 sin( 2 3 ϕ) is the exact solution of the problem. The solution belongs to V H 5/3 ε (Ω) for any ε > 0 (but not to H 5/3 (Ω)!) and has a singularity in the origin.

7 A Posteriori rror stimation V Finite element approximations are based on the Ritz-Galerkin approach: Given a finite dimensional subspace V h V of test/trial functions, find u h V h such that a(u h,v h ) = l(v h ), v h V h. Since V h V, the existence and uniqueness of a discrete solution u h V h follows readily from the Lax-Milgram Lemma. Moreover, we deduce that the error e u := u u h satisfies the Galerkin orthogonality a(u u h,v h ) = 0, v h V h, i.e., the approximate solution u h V h is the projection of the solution u V onto V h with respect to the inner product a(, ) on V (elliptic projection). Using the Galerkin orthogonality, it is easy to derive the a priori error estimate u u h 1,Ω M inf v h V h u v h 1,Ω, where M := C/γ. This result tells us that the error is of the same order as the best approximation of the solution u V by functions from the finite dimensional subspace V h. It is known as Céa s Lemma.

8 A Posteriori rror stimation VI The Ritz-Galerkin method also gives rise to an a posteriori error estimate in terms of the residual r : V R In fact, it follows that for any v V r(v) := l(v) a(u h,v), v V. γ u u h 2 1,Ω a(u u h,u u h ) = r(u u h ) r 1,Ω u u h 1,Ω, whence u u h 1,Ω 1 γ sup r(v). v V v 1,Ω

9 Department of Mathematics, University of Houston Reliability and fficiency

10 Reliability and fficiency of rror stimators I Definition. An error estimator η h is called reliable, if it provides an upper bound for the error up to data oscillations osc rel h, i.e., if there exists a constant C rel > 0, independent of the mesh size h of the underlying triangulation, such that e u a C rel η h + osc rel h. On the other hand, an estimator η h is said to be efficient, if up to data oscillations osc eff h it gives rise to a lower bound for the error, i.e., if there exists a constant C eff > 0, independent of the mesh size h of the underlying triangulation, such that η h C eff e u a + osc eff h. Finally, an estimator η h is called asymptotically exact, if it is both reliable and efficient with C rel = C 1 eff.

11 Reliability and fficiency of rror stimators II Remark. The notion reliability is motivated by the use of the error estimator in error control. Given a tolerance tol, an idealized termination criterion would be e u a tol. Since the error e u a is unknown, we replace it with the upper bound, i.e., C rel η h + osc rel h tol. We note that the termination criterion both requires the knowledge of C rel and the incorporation of the data oscillation term osc rel and osc rel h 0, it reduces to η h tol. h. In the special case C rel = 1 An alternative, but less used termination criterion is based on the lower bound, i.e., we require 1 ( ) η h osc eff h tol. C eff Typically, this criterion leads to less refinement and thus requires less computational time which motivates to call the estimator efficient.

12 Department of Mathematics, University of Houston Residual-Type A Posteriori rror stimation

13 The error estimate The Role of the Residual u u h 1,Ω 1 γ sup r(v) v V v 1,Ω shows that in order to assess the error e u a we are supposed to evaluate the norm of the residual with respect to the dual space V, i.e., r V := r(v) sup. v V\{0} v a In particular, we have the equality r V = e u a, whereas for the relative error of r(v),v V, as an approximation of e u a we obtain ( e u a r(v)) = 1 e u a 2 v e u 2 e u a, v V with v a = 1. a The goal is to obtain lower and upper bounds for r V at relatively low computational expense.

14 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 Lu := (a u) = f in Ω, u = 0 on Γ D, n a u = g on Γ N, where f L 2 (Ω), g L 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 and uniformly positive definite. The vector n denotes the exterior unit normal vector on Γ N. Setting H 1 0,Γ D (Ω) := { v H 1 (Ω) v ΓD = 0 }, the weak formulation is as follows: Find u H 1 0,Γ D (Ω) such that where a(v,w) := Ω a(u,v) = l(v), v H 1 0,Γ D (Ω), a v w dx, l(v) := f v dx + Ω Γ N g v dσ, v H 1 0,Γ D (Ω).

15 F Approximation: 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 T P 1 (K), T T h } the trial space of continuous, piecewise linear finite elements with respect to T h. Note that P k (T), k 0, denotes the linear space of polynomials of degree k on T. 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. We further denote by T the area, by h T the diameter of an element T T h, and by h = the length of an edge h (Ω Γ N ). We refer to f T := T 1 T fdx the integral mean of f with respect to an element T T h and to g := 1 gds the mean of g with respect to the edge h(γ N ). The conforming P1 approximation reads as follows: Find u h S 1,ΓD (Ω; T h ) such that a(u h,v h ) = l(v h ), v h S 1,ΓD (Ω; T h ).

16 The residual r is given by r(v) := f v dx + Representation of the Residual I Ω Γ N g vds a(u h,v), v V. Applying Green s formula elementwise yields a(u h,v) = a u h v dx = [n a u h ] v ds + n a u h v ds, T T h T h (Ω) h (Γ N ) where [n a u h ] denotes the jump of the normal derivative of u h across h (Ω) and where we have used that u h 0 on T T h, since u h T P 1 (T). We thus obtain r(v) := r T (v) + r (v). T T h h (Ω Γ N )

17 Department of Mathematics, University of Houston Representation of the Residual II Here, the local residuals r T (v),t T h, are given by r T (v) := (f Lu h )v dx, whereas for r (v) we have r (v) := r (v) := T [n a u h ]v ds, h (Ω), (g n a u h ) v ds, h (Γ N ).

18 A Posteriori rror stimator and Data Oscillations The error estimator η h consists of element residuals η T,T T h, and edge residuals η, H (Ω Γ N ), according ( to 1/2, η h := ηt 2 + η) 2 T T h H (Ω Γ N ) where η T and η are given by η T := h T f T Lu h 0,T, T T h, { h 1/2 η := [n a u h] 0,, h (Ω), h 1/2 g n a u h 0,, h (Γ N ). The a posteriori error analysis ( further invokes the data oscillations 1/2, osc h := osc 2 T(f) + osc(g)) 2 T T h h (Γ N ) where osc T (f) and osc (g) are given by osc T (f) := h T f f T 0,T, osc (g) := h 1/2 g g 0,.

19 Clément s Quasi-Interpolation Operator I 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 D p := { T T h p N h (T) }. 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 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 )

20 Clément s Quasi-Interpolation Operator II Theorem. Let v H 1 0,Γ D (Ω). Then, for Clément s interpolation operator there holds P C v 0,T C v 0,D (1) T v P C v 0,T C h T v 1,D (1) T Further, we have ( ( h (Ω) h (Γ N ), P C v 0, C v (1) 0,D, v P C v 0, C h 1/2 v 2 µ,d (1) K T T h v 2 µ,d (1), P C v 0,T C v (1) 0,D, T. v 1,D (1) ) 1/2 C v µ,ω, 0 µ 1, ) 1/2 C v µ,ω, 0 µ 1. where D (1) T := { T T h N h (T ) N h (T) }, D (1) := { T T h N h () N h (T ) }.

21 Reliability of the A posteriori rror stimator I Theorem. There exist constants Γ R and Γ osc > 0 depending only on the shape regularity of T h such that e u a Γ R η h + Γ osc osc h. Proof. Setting v = e u, we have e u a = a(e u,e u ) = r(e u ) = r(p C e u ) + r(e u P C e u ). For the first term on the right-hand side, by Galerkin orthogonality we obtain r(p C e u ) = f P C e u dx + g P C e u ds a(u h,p C e u ) = 0. Γ N Ω

22 Proof cont d. On the other hand, for the second term on the right-hand side of Green s formula yields r(e u P C e u ) = f (e u P C e u ) dx + g (e u P C e u ) ds + } a {{ u } h (e u P C e u ) dx Ω Γ T T h N T = Lu h n T a u h (e u P C e u ) ds = (f T Lu h ) (e u P C e u ) dx T T h T T h + T h (Ω) + T T h T [n a u h ] (e u P C e u ) ds + (f f T ) (e u P C e u ) dx + h (Γ N ) T h (Γ N ) (g n a u h ) (e u P C e u ) ds (g g ) (e u P C e u ) ds.

23 Proof cont d. In view of the local approximation properties of Clément s quasiinterpolation operator, it follows that ( r(e u P C e u ) C h 2 T f T Lu h 2 0,T) 1/2 e u 1,Ω T T h + h [n a u h ] J 2 0,) 1/2 e u 1,Ω + + h (Ω) h (Γ N ) h (Γ N ) h g n a u h 2 0,) 1/2 e u 1,Ω + ( T T h h 2 T f f T 2 0,T) 1/2 e u 1,Ω h g g 2 0,) 1/2 e u 1,Ω ), from which we may conclude.

24 lement and dge Bubble Functions I The element bubble function ψ T is defined by means of the barycentric coordinates λ T i,1 i 3, according to ψ T := 27 λ T 1 λ T 2 λ T 3. Note that supp ψ T = T int, i.e., ψ T T = 0, T T h. On the other hand, for h (Ω) h (Γ N ) and T T h such that T and p i N h (), 1 i 2, we introduce the edge-bubble functions ψ ψ := 4 λ T 1 λ T 2. Note that ψ = 0 for h (T),.

25 lement and dge Bubble Functions II The bubble functions ψ T and ψ have the following important properties that can be easily verified taking advantage of the affine equivalence of the finite elements: Lemma. There holds p h 2 0,T C p h 2 0, C T p 2 h ψ T dx, p 2 h ψ dσ, p h P 1 (T), p h P 1 (), p h ψ T 1,T C h 1 T p h 0,T, p h P 1 (T), p h ψ T 0,T C p h 0,T, p h P 1 (T), p h ψ 0, C p h 0,, p h P 1 ().

26 lement and dge Bubble Functions III For functions p h P 1 (), h (Ω) h (Γ N ) we further need an extension p h L2 (T) where T T h such that T. For this purpose we fix some T,, and for x T denote by x that point on such that (x x ). For p h P 1 () we then set p h := p h (x ). as the union of elements T T h con- Further, for h (Ω) h (Γ N ) we define D (2) taining as a common edge D (2) := { K T h h (T) }.

27 Department of Mathematics, University of Houston Lemma. There holds lement and dge Bubble Functions IV p h ψ (2) 1,D p h ψ (2) 0,D C h 1/2 p h 0,e, p h P 1 (), C h 1/2 p h 0,, p h P 1 (). Further, for all v V and µ = 0,1 there holds ( h 1 µ ) 1/2 C ( h 1 µ T v 2 µ,t) 1/2. T T h h (Ω) h (Γ N ) v 2 µ,d (2)

28 fficiency of the A posteriori rror stimator I Theorem. There exist constants γ R,γ > 0, depending only on the shape regularity of T h such that γ R η h γ osc h e 1,Ω. Proof. The proof will be given by a series of lemmas establishing the local efficiency of the estimator.

29 Lemma. Let T T h. Then there holds: h T f T Lu h 0,T C e 1,T + h T f f T 0,T. Proof. We set p h := f T. Observing ψ T T = 0, by Green s formula a T (u h,p h ψ T ) = (a u h ) p h ψ T dx + n } T a u {{ h p h ψ T} = 0 T Denoting by π h ψ T the L 2 -projection onto the linear space of elementwise constants and taking advantage of the properties of the bubble functions, it follows that ( f T Lu h 2 0,T C (f T Lu h ) π h ψ T dx = f π h ψ T dx a T (u h, π h ψ T ) + T ) (π h f f) π h ψ T dx = ( a T (e u,π h ψ T ) + T T ) (f T f) π h ψ T dx ds. T C h 1 T e u 1,T p h 0,T + f T f 0,T p h 0,T, T

30 Department of Mathematics, University of Houston Lemma. Let h (Ω). Then there holds: h 1/2 [n a u h ] 0, C e (2) 1,D + h f f T (2) 0,D D (2) D (2) + h f T Lu h (2) 0,D. Proof. We set p h := [n a u h ]. In view of ψ = 0,,, Green s formula gives n (2) D a u h p h ψ ds = a (2) D (u h,p h ψ ) + a u h }{{} p h ψ dx, = Lu h where p h ψ is the L 2 -projection onto the edgewise constants.

31 Proof cont d. If we use the properties of the bubble functions, it follows that [n (2) D a u h ] 2 0, C a u h ] p h ψ ds = a u h ] p h ψ ds = ( a D (2) (u h,p h ψ ) = a (2) D (e,p h ψ ) + C h 1/2 e u 1,D (2) D (2) ( D (2) [n D (2) f p h ψ dx + D (2) (f f T ) p h ψ dx + p h 0, + h 1/2 f f T 0,D (2) D (2) [n D (2) (f f T ) p h ψ dx + D (2) from which the assertion can be easily deduced. D (2) ) (f T Lu h ) p h ψ dx ) (f T Lu h ) p h ψ dx p h 0, + h 1/2 f T Lu h 0,D (2) p h 0,.

32 Lemma. Let h (Γ N ). Then there holds: h 1/2 g n u h 0, C e (2) 1,D + h 1/2 g g 0, + h f f T 0,D (2) + h f T Lu h (2) 0,D. Proof. We set p h := g n a u h. Observing ψ = 0,, by Green s formula we obtain n a u h p h ψ ds = a u h p h ψ ds = a (2) D (u h,p h ψ ) + D (2) D (2) n D (2) a u h }{{} = Lu h p h ψ dx.

33 Department of Mathematics, University of Houston Goal-Oriented Dual Weighted Approach

34 Goal-Oriented Dual Weighted Approach I The goal oriented dual weighted approach allows to control the error e u := u u h with respect to a rather general error functional or output functional J : V H 1 (Ω) lr. The goal oriented dual weighted approach strongly uses the solution z V of the associated dual problem a(v,z) = J(v), v V, and its finite element approximation z h V h, i.e., a(v h,z h ) = J(v h ), v h V h. Using Galerkin orthogonality, we readily deduce that J(e u ) = a(e u,z) = a(e u,z v h ) = r(z v h ), v h V h, where r( ) stands for the residual with respect to the computed finite element approximation u h.

35 Goal-Oriented Dual Weighted Approach II Theorem. Let u h V h := S 1,Γ (Ω; T h (Ω)) be the conforming P1 approximation of the solution u H 1 0(Ω) of Poisson s equation with f L 2 (Ω) and homogeneous Dirichlet boundary data. Then, the following error representation holds true J(e u ) = ) ((r T,z v h ) 0,T + (r T,z v h ) 0, T, v h V h, T T h (Ω) where the element residuals r T and the edges residuals r T are given by { 1 r T := f, T T h (Ω), r T := 2 ν [ u h ], h ( T Ω) 0, h ( T Γ) Moreover, we have the error estimate J(e u ) η DW := ω T ρ T, T T h (Ω) where for v h V h the element residuals ρ T and the weights ω T read ( 1/2, ( 1/2. ρ T := r T 2 0,T + h 1 T r T 0, T) 2 ωt := z v h 2 0,T + h T z v h 0, T) 2

36 Department of Mathematics, University of Houston Goal-Oriented Dual Weighted Approach III We remark that the previous result is not really a posteriori, since the solution z V of the dual solution is not known. Therefore, information about the weights ω T,T T h (Ω) has to be provided either by means of an a priori analysis or by the numerical solution of the dual problem. Theorem. Under the assumptions of the previous theorem let the error functional be given by Then, there holds J(v) := ( v, e u) 0,Ω e u 0,Ω, v V. e u 0,Ω C ( T T h (Ω) h 2 T ρ 2 T) 1/2.

37 Proof. The dual solution z V satisfies a(v,z) = ( v, e u) 0,Ω e u 0,Ω, v V, from which we readily deduce the a priori bound z 0,Ω 1. In view of the basic error estimate it follows that ( ) 1/2 ( J(e u ) = e u 0,Ω h 2 T ρ 2 T T T h (Ω) T T h (Ω) h 2 T ω2 T) 1/2. Choosing v h = P C z, where P C is Clément s quasi-interpolation operator, we find ( 1/2 inf (h 2 T z v h 2 0,T + h 1 T z v h 0, T) 2 C z 0,Ω. v h V h T T h (Ω) Using the last inequality in the previous one and observing the error representation gives the assertion.

38 Goal-Oriented Dual Weighted Approach IV Theorem. Consider the conforming P1 approximation of Poisson s equation under homogeneous Dirichlet boundary conditions and assume that the solution u V := H 1 0(Ω) is 2-regular. Using the the error functional J(v) := (v,e u) 0,Ω e u 0,Ω, v V, gives rise to the a posteriori error estimate ( e u 0,Ω C T T h (Ω) h 4 T ρ 2 T) 1/2.

39 Goal-Oriented Dual Weighted Approach V Finally, we apply the goal-oriented dual weighted approach to the pointwise estimation of the error at some point a Ω. Given some tolerance ε > 0, we consider the ball K ε (a) := {x Ω x a < ε} around the point a and define the regularized error functional J(v) := Kε(a) 1 v dx. Kε (a) The dual solution z of a(v,z) = J(v) behaves like a regularized Green s function With the residual ρ T we obtain z(x) log(r(x)), r(x) := x a 2 + ε 2. (u u h )(a) T T h (Ω) h 3 T r 2 T ρ T, r T := max x T r(x).