Approximation of fluid-structure interaction problems with Lagrange multiplier

Transcription

1 Approximation of fluid-structure interaction problems with Lagrange multiplier Daniele Boffi Dipartimento di Matematica F. Casorati, Università di Pavia May 30, 2016

2 Outline 1 An interface problem 2 Fluid-Structure Interaction 3 Lagrange multiplier 4 Saddle point problem Main collaborators: Lucia Gastaldi, Nicola Cavallini, Michele Ruggeri

3 Outline 1 An interface problem 2 Fluid-Structure Interaction 3 Lagrange multiplier 4 Saddle point problem

4 An interface problem Let us consider a standard interface problem Auricchio B. Gastaldi Lefieux Reali 13 B. Gastaldi Ruggeri 13 div(β 1 u 1 ) = f 1 in Ω 1 div(β 2 u 2 ) = f 2 in Ω 2 u 1 = u 2 on Γ β 1 u 1 n 1 + β 2 u 2 n 2 = 0 on Γ u 1 = 0 on Ω 1 \ Γ u 2 = 0 on Ω 2 \ Γ Daniele Boffi FD-DLM-IBM May 30, 2016 page 3

5 Mixed formulation Notation: Ω = Ω 1 Ω 2 Find u H0 1(Ω), u 2 H 1 (Ω 2 ), and λ Λ = [H 1 (Ω 2 )] such that β u v dx + λ, v Ω2 = fv dx Ω Ω v H0(Ω) 1 (β 2 β) u 2 v 2 dx λ, v 2 = Ω 2 (f 2 f)v 2 dx Ω 2 v 2 H 1 (Ω 2 ) µ, u Ω2 u 2 = 0 µ Λ Equivalent to interface problem if β Ω1 = β 1 and f Ω1 = f 1 We get u Ω1 = u 1 Daniele Boffi FD-DLM-IBM May 30, 2016 page 4

6 Alternative mixed formulation Find u H0 1(Ω), u 2 H 1 (Ω 2 ), and ψ H 1 (Ω 2 ) such that β u v dx + ((ψ, v Ω2 )) = fv dx Ω Ω v H0(Ω) 1 (β 2 β) u 2 v 2 dx ((ψ, v 2 )) = Ω 2 (f 2 f)v 2 dx Ω 2 v 2 H 1 (Ω 2 ) ((ϕ, u Ω2 u 2 )) = 0 ϕ H 1 (Ω 2 ) where ((, )) denotes the scalar product in H 1 (Ω 2 ) Daniele Boffi FD-DLM-IBM May 30, 2016 page 5

7 Alternative mixed formulation Find u H0 1(Ω), u 2 H 1 (Ω 2 ), and ψ H 1 (Ω 2 ) such that β u v dx + ((ψ, v Ω2 )) = fv dx v H0(Ω) 1 Ω Ω (β 2 β) u 2 v 2 dx ((ψ, v 2 )) = (f 2 f)v 2 dx v 2 H 1 (Ω 2 ) Ω 2 Ω 2 ((ϕ, u Ω2 u 2 )) = 0 ϕ H 1 (Ω 2 ) where ((, )) denotes the scalar product in H 1 (Ω 2 ) Remark The two mixed formulations are equivalent but give rise to different discrete schemes Daniele Boffi FD-DLM-IBM May 30, 2016 page 5

8 Approximation of mixed formulations Two meshes: T h for Ω and T 2,h for Ω 2 Three finite element spaces: V h continuous p/w linears on T h V 2,h continuous p/w linears on T 2,h Λ h = V 2,h Daniele Boffi FD-DLM-IBM May 30, 2016 page 6

9 Approximation of mixed formulations Two meshes: T h for Ω and T 2,h for Ω 2 Three finite element spaces: V h continuous p/w linears on T h V 2,h continuous p/w linears on T 2,h Λ h = V 2,h Several other choices are possible Remark First mixed formulation makes use of V h, V 2,h, and Λ h (duality represented by scalar product in L 2 (Ω 2 )) Second mixed formulation makes use of V h, V 2,h, and V 2,h Daniele Boffi FD-DLM-IBM May 30, 2016 page 6

10 Stability of the approximation We need to show the ellipticity in the kernel and the inf-sup condition ELKER ( ) β v 2 dx + Ω (β 2 β) v 2 2 dx κ 1 Ω 2 v 2 H 1 (Ω) + v 2 2 H 1 (Ω 2 ) (v, v 2 ) K h where the kernel K h is defined as K h = {(v, v 2 ) V h V 2,h : (µ, v Ω2 v 2 ) = 0 µ Λ h } or, for the second formulation, K h = {(v, v 2 ) V h V 2,h : ((ϕ, v Ω2 v 2 )) = 0 ϕ V 2,h } Daniele Boffi FD-DLM-IBM May 30, 2016 page 7

11 Stability of the approximation (cont ed) INFSUP1 (v,v 2 ) V h Λ h v 2 H 1 (Ω) + v 2 2 H 1 (Ω 2 ) sup ( (µ, v Ω2 v 2 ) ) 1/2 κ 2 µ Λ µ Λ h INFSUP2 ((ϕ, v Ω2 v 2 )) sup (v,v 2 ) V h V 2,h ( v 2 H 1 (Ω) + v 2 2 κ 2 ϕ H H 1 (Ω 2 ) )1/2 1 (Ω 2 ) ϕ V 2,h Daniele Boffi FD-DLM-IBM May 30, 2016 page 8

12 Stability of the approximation (cont ed) Theorem If β 2 β Ω2 η 0 > 0 then ELKER holds true for both formulations, uniformly in h and h 2 Remark For the second mixed formulation, ELKER holds true without assumptions on β if h 2 /h d/2 is small enough and T h is quasi-uniform Daniele Boffi FD-DLM-IBM May 30, 2016 page 9

13 Stability of the approximation (cont ed) Theorem If the mesh sequence T 2,h is quasi-uniform, then INFSUP1 holds true, uniformly in h and h 2 Theorem INFSUP2 holds true, uniformly in h and h 2 without any additional assumptions on the mesh sequence Daniele Boffi FD-DLM-IBM May 30, 2016 page 10

14 Some numerical examples Let us consider a one dimensional example where the interval (0, 6) is subdivided into (0, e) (e, 1 + π) (1 + π, 6) β 1 β 2 β 1 We compare the standard Galerkin method and our method in its two variants (all finite element spaces are continuous p/w linears) Daniele Boffi FD-DLM-IBM May 30, 2016 page 11

15 Standard Galerkin method β 1 = 1, β 2 = 10 β 1 = 10, β 2 = 1 Daniele Boffi FD-DLM-IBM May 30, 2016 page 12

16 Standard Galerkin method β 1 = 1, β 2 = β 1 = 10000, β 2 = 1 Daniele Boffi FD-DLM-IBM May 30, 2016 page 13

17 DLM/L2 β 1 = 1, β 2 = 10 Daniele Boffi FD-DLM-IBM May 30, 2016 page 14

18 DLM/H1 β 1 = 1, β 2 = 10 Daniele Boffi FD-DLM-IBM May 30, 2016 page 15

19 DLM/L2 β 1 = 1, β 2 = Daniele Boffi FD-DLM-IBM May 30, 2016 page 16

20 DLM/H1 β 1 = 1, β 2 = Daniele Boffi FD-DLM-IBM May 30, 2016 page 17

21 DLM/L2 β 1 = 10, β 2 = 1 Daniele Boffi FD-DLM-IBM May 30, 2016 page 18

22 DLM/H1 β 1 = 10, β 2 = 1 Daniele Boffi FD-DLM-IBM May 30, 2016 page 19

23 DLM/L2 β 1 = 10000, β 2 = 1 Daniele Boffi FD-DLM-IBM May 30, 2016 page 20

24 DLM/H1 β 1 = 10000, β 2 = 1 Daniele Boffi FD-DLM-IBM May 30, 2016 page 21

25 Outline 1 An interface problem 2 Fluid-Structure Interaction 3 Lagrange multiplier 4 Saddle point problem

26 Fluid-structure interaction Ω R d, d = 2, 3 x Euler. var. in Ω B t deformable structure domain B t R m, m = d, d 1 s Lagrangian var. in B X(, t) : B B t position of the solid F = X deformation gradient s Ω B t B X Daniele Boffi FD-DLM-IBM May 30, 2016 page 23

27 Fluid-structure interaction Ω R d, d = 2, 3 x Euler. var. in Ω B t deformable structure domain B t R m, m = d, d 1 s Lagrangian var. in B X(, t) : B B t position of the solid F = X deformation gradient s Ω B t B X u(x, t) material velocity u(x, t) = X (s, t) where x = X(s, t) t Daniele Boffi FD-DLM-IBM May 30, 2016 page 23

28 Model assumptions From conservation of momenta, in absence of external forces, ( ) u ρ u = ρ t + u u = div σ in Ω Structural material with density ρ s different from fluid density ρ f { ρ f in Ω \ B t ρ = ρ s in B t In our case the Cauchy stress tensor has the following form { σ f in Ω \ B t σ = σ f + σ s in B t Daniele Boffi FD-DLM-IBM May 30, 2016 page 24

29 Model assumptions (cont ed) Incompressible fluid: σ f = pi + µ sym u where sym = u + ( u) Visco-elastic incompressible material: σ = σ f + σ s with σ s elastic part of the stress The Piola Kirchhoff stress tensor takes into account the change of variable P = F σ s F and P(F) = W F where W is the potential energy density Daniele Boffi FD-DLM-IBM May 30, 2016 page 25

30 FSI problem (thick solid) ( ) uf ρ f t + u f u f = div σ f f in Ω \ B t div u f = 0 in Ω \ B t 2 X ρ s t 2 = div s( F σ s f F + P(F)) in B div s u s = 0 in B u f = X t on B t σ f f n f = (σ s f + F 1 PF )n s on B t σ f f = p f I + µ sym u f σ s f = p si + µ sym u s u s = X t + initial and boundary conditions Daniele Boffi FD-DLM-IBM May 30, 2016 page 26

31 Outline 1 An interface problem 2 Fluid-Structure Interaction 3 Lagrange multiplier 4 Saddle point problem

32 FSI with Lagrange multiplier Fluid velocity and pressure are extended into the solid domain { { uf in Ω \ B u = t pf in Ω \ B p = t u s in B t p s in B t Daniele Boffi FD-DLM-IBM May 30, 2016 page 28

33 FSI with Lagrange multiplier Fluid velocity and pressure are extended into the solid domain { { uf in Ω \ B u = t pf in Ω \ B p = t u s in B t p s in B t Body motion u(x, t) = X (s, t) for x = X(s, t) t Daniele Boffi FD-DLM-IBM May 30, 2016 page 28

34 FSI with Lagrange multiplier Fluid velocity and pressure are extended into the solid domain { { uf in Ω \ B u = t pf in Ω \ B p = t u s in B t p s in B t Body motion u(x, t) = X (s, t) for x = X(s, t) t We introduce two functional spaces Λ and Z and a bilinear form c : Λ Z R such that c(µ, Z) = 0 µ Λ Z = 0 The definition of Λ, Z and c depends on the codimension of the immersed solid. For simplicity, we take P = κf = κ s X Daniele Boffi FD-DLM-IBM May 30, 2016 page 28

35 FE-IBM with DLM - Variational formulation Notation: B. Cavallini Gastaldi 14 { νf in Ω a(u, v) = (ν sym u, sym v) with ν = f ν s in Ω s b(u, v, w) = ρ f 2 ((u v, w) (u w, v)) δρ = ρ s ρ f Daniele Boffi FD-DLM-IBM May 30, 2016 page 29

36 Variational formulation (cont ed) For t [0, T], find u(t) H 1 0 (Ω)d, p(t) L 2 0 (Ω), X(t) W1, (B) d, and λ(t) Λ such that ρ d (u(t), v) + a(u(t), v) + b(u(t), u(t), v) dt (div v, p(t)) + c(λ, v(x(, t))) = 0 v H 1 0(Ω) d (div u(t), q) = 0 q L 2 0(Ω) 2 X δρ B t 2 Yds + κ s X s Y ds c(λ, Y) = 0 B Y H 1 (B) d ( c µ, u(x(, t), t) X(t) ) = 0 t µ Λ Daniele Boffi FD-DLM-IBM May 30, 2016 page 30

37 Time advancing scheme Modified backward Euler Given u 0 H0 1(Ω)d and X 0 W 1, (B) d, for n = 1,..., N, find (u n, p n ) H0 1(Ω)d L 2 0 (Ω), Xn W 1, (B) d, and λ n Λ, such that ( u n+1 u n ) ρ f, v + a(u n+1, v) + b(u n+1, u n+1, v) t (div u n+1, q) = 0 (div v, p n+1 ) + c(λ n+1, v(x n ( ))) = 0 ( X n+1 2X n + X n 1 ) δρ t 2, Y + κ( s X n+1, s Y) B B c(λ n+1, Y) = 0 v H 1 0(Ω) d q L 2 0(Ω) Y H 1 (B) c (µ, u n+1 (X n ( )) Xn+1 X n ) = 0 µ Λ t Daniele Boffi FD-DLM-IBM May 30, 2016 page 31

38 Energy estimate for the time discrete problem Proposition (Unconditional stability) Assume that W is convex and δρ = ρ s ρ f > 0 The following estimate holds true for all n = 1,..., N ρ f ( u n u n 2 ) 0 + µ u n t ( + δρ X n+1 X n 2 2 t t X n X n 1 t 0,B + 1 t (E(Xn+1 ) E(X n )) 0 where E(X) is the elastic potential energy given by E(X) = κ s X(s) 2 ds 2 B 2 0,B ) Daniele Boffi FD-DLM-IBM May 30, 2016 page 32

39 Outline 1 An interface problem 2 Fluid-Structure Interaction 3 Lagrange multiplier 4 Saddle point problem

40 Stationary problem Let X W 1, (B) d be invertible with Lipschitz inverse. Given f L 2 (Ω) d, g L 2 (B) d and d L 2 (B) d, find u H 1 0 (Ω)d, p L 2 0 (Ω), X H 1 (B) d and λ Λ such that a f (u, v) (div v, p) + c(λ, v(x( ))) = (f, v) (div u, q) = 0 a s (X, Y) c(λ, Y) = (g, Y) B c(µ, u(x( )) X) = c(µ, d) v H0(Ω) 1 d q L 2 0(Ω) Y H 1 (B) d µ Λ where a f (u, v) = α(u, v) + a(u, v) a s (X, Y) = β(x, Y) B + κ( s X, s Y) B u, v H 1 0(Ω) d X, Y H 1 (B) d Daniele Boffi FD-DLM-IBM May 30, 2016 page 34

41 Choice of c Thick immersed solid (codimension zero) Notice that X W 1, (B) d implies v(x( )) H 1 (B) d Case 1. Z = H 1 (B) d, Λ dual space of H 1 (B) d,, duality pairing c(λ, Z) = λ, Z λ Λ = (H 1 (B)), Z H 1 (B) d Case 2. Z = H 1 (B) d, Λ = H 1 (B) d c(λ, Z) = ( s λ s Z + λ Z) ds B λ Λ, Z H 1 (B) d Daniele Boffi FD-DLM-IBM May 30, 2016 page 35

42 Choice of c (cont ed) Thin immersed solid (codimension one) In this case v(x( )) is the trace of v and it belongs to H 1/2 (B) d We set Z = H 1/2 (B) d, Λ dual space of H 1/2 (B) d,, duality pairing c(λ, Z) = λ, Z λ Λ = (H 1/2 (B) d ), Z H 1/2 (B) d Daniele Boffi FD-DLM-IBM May 30, 2016 page 36

43 Operator form of the problem The problem has the following double saddle point structure A B 0 C 1 B A s C 2 C 1 0 C 2 0 U P X Λ = F 0 G D Daniele Boffi FD-DLM-IBM May 30, 2016 page 37

44 Well-posedness of the saddle point problem - I V = H 1 0 (Ω)d L 2 0 (Ω) H1 (B) d K = { (v, q, Y) V : c(µ, v(x( )) Y) = 0 µ Λ } K = {(v, q, Y) V : v(x( )) = Y} First inf-sup condition There exists α 0 > 0 such that inf (u,p,x) K a f (u, v) (p, div v) + (div u, q) + a s (X, Y) sup α 0. (v,q,y) K (u, p, X) V (v, q, Y) V Daniele Boffi FD-DLM-IBM May 30, 2016 page 38

45 Well-posedness of the saddle point problem - I V = H 1 0 (Ω)d L 2 0 (Ω) H1 (B) d K = { (v, q, Y) V : c(µ, v(x( )) Y) = 0 µ Λ } K = {(v, q, Y) V : v(x( )) = Y} First inf-sup condition There exists α 0 > 0 such that inf (u,p,x) K a f (u, v) (p, div v) + (div u, q) + a s (X, Y) sup α 0. (v,q,y) K (u, p, X) V (v, q, Y) V Proof Consequence of well-posedness of mixed Navier Stokes and ellipticity of a s Daniele Boffi FD-DLM-IBM May 30, 2016 page 38

46 Well-posedness of the saddle point problem - II Inf-sup condition There exists β 0 > 0 such that c(µ, v(x( )) Y) sup β 0 µ Λ (v,q,y) V (v, q, Y) V Daniele Boffi FD-DLM-IBM May 30, 2016 page 39

47 Well-posedness of the saddle point problem - II Inf-sup condition There exists β 0 > 0 such that c(µ, v(x( )) Y) sup β 0 µ Λ (v,q,y) V (v, q, Y) V Thick immersed solid The proof depends on the choice of c, e.g. in Case 1 µ Λ = µ, Y c(µ, v(x( )) Y) sup sup Y H 1 (B) d Y H 1 (B) (v,q,y) V (v, q, Y) V Daniele Boffi FD-DLM-IBM May 30, 2016 page 39

48 Well-posedness of the saddle point problem - II Inf-sup condition There exists β 0 > 0 such that c(µ, v(x( )) Y) sup β 0 µ Λ (v,q,y) V (v, q, Y) V Thick immersed solid The proof depends on the choice of c, e.g. in Case 1 µ Λ = Thin immersed solid µ, Y c(µ, v(x( )) Y) sup sup Y H 1 (B) d Y H 1 (B) (v,q,y) V (v, q, Y) V µ Λ = µ, z sup z H 1/2 (B) z d H 1/2 (B) We construct a maximizing sequence z n H 1/2 (B) d and corresponding functions v n H 1 0(Ω) d such that v n(x( )) = z n Daniele Boffi FD-DLM-IBM May 30, 2016 page 39

49 Finite element discretization We consider Background grid T h for the domain Ω (meshsize h x ) (V h, Q h ) H0 1(Ω)d L 2 0 (Ω) stable pair for the Stokes equations Daniele Boffi FD-DLM-IBM May 30, 2016 page 40

50 Finite element discretization We consider Background grid T h for the domain Ω (meshsize h x ) (V h, Q h ) H0 1(Ω)d L 2 0 (Ω) stable pair for the Stokes equations Grid S h for B (meshsize h s ) S h H 1 (B) continuous Lagrange elements S h = {Y C 0 (B; Ω) : Y P 1 } Daniele Boffi FD-DLM-IBM May 30, 2016 page 40

51 Finite element discretization We consider Background grid T h for the domain Ω (meshsize h x ) (V h, Q h ) H0 1(Ω)d L 2 0 (Ω) stable pair for the Stokes equations Grid S h for B (meshsize h s ) S h H 1 (B) continuous Lagrange elements S h = {Y C 0 (B; Ω) : Y P 1 } Λ h Λ continuous Lagrange elements. We take Λ h = S h Daniele Boffi FD-DLM-IBM May 30, 2016 page 40

52 Discrete saddle point problem Find u h V h, p h Q h, X h S h and λ h Λ h such that a f (u h, v) (div v, p h ) + c(λ h, v(x( ))) = (f, v) (div u h, q) = 0 a s (X h, Y) c(λ h, Y) = (g, Y) B c(µ, u h (X( )) X h ) = c(µ, d) v V h q Q h Y S h µ Λ h Daniele Boffi FD-DLM-IBM May 30, 2016 page 41

53 Analysis of the discrete problem V h = V h Q h S h K h = { (v, q, Y) V h : c(µ, v(x( )) Y) = 0 µ Λ h } First discrete inf-sup condition Assume that V h Q h is stable for Stokes, then there exists α 1 > 0 independent of h x and h s such that inf (u,p,x) K h a f (u, v) (p, div v) + (div u, q) + a s (X, Y) sup α 1 (v,q,y) K h (u, p, X) V (v, q, Y) V Proof Consequence of discrete stability of Stokes element and ellipticity of a s Daniele Boffi FD-DLM-IBM May 30, 2016 page 42

54 Second discrete inf-sup condition I Thick immersed solid Case 1 For all µ Λ h = S h and Z H 1 (B) c(µ, Z) = B µzds. Let P 0 : H 1 (B) S h be the L 2 -projection We assume that the mesh sequence S h is such that P 0 is H 1 -stable P 0 v 1,B c v 1,B v H 1 (B) then there exists β 1 > 0 independent of h x and h s such that c(µ, v(x( )) Y) sup β 1 µ (v,q,y) V (v, q, Y) (H 1 (B)) V Daniele Boffi FD-DLM-IBM May 30, 2016 page 43

55 Sufficient conditions for H 1 -stability of P 0 S h quasi-uniform, use the inverse inequality neighboring element-sizes obeys a global growth-condition Crouzeix Thomée 87, weakened in Carstensen 02 mesh locally quasi-uniform Bramble Pasciak Steinbach 02 the diameter h(t) of an element T satisfies α2 k(t) h(t) β2 k(t), where k(t) is the level of the element and the level of two neighboring elements differs at most by one Bank Yserentant 14 Daniele Boffi FD-DLM-IBM May 30, 2016 page 44

56 Second discrete inf-sup condition II Thick immersed solid Case 2 c(µ, Z) = ( s µ s Z + µz)ds B µ Λ, Z H 1 (B) d There exists β 1 > 0 independent of h x and h s such that c(µ, v(x( )) Y) sup β 1 µ (v,q,y) V (v, q, Y) H 1 (B) V The proof is the same as in the continuous case. Daniele Boffi FD-DLM-IBM May 30, 2016 page 45

57 Second discrete inf-sup condition Thin immersed solid Recall Λ = (H 1/2 (B) d ), c : Λ H 1/2 (B) d R and c(µ, Z) = µ, Z. S h = {Y H 1 (B) d : Y piecewise linear on S h } Λ h = S h For all µ Λ h, v V h and Y S h c(µ, v(x( )) Y) = µ(v(x( )) Y)ds B If the mesh is quasi-uniform and h x /h s is sufficiently small, there exists β 2 > 0 independent of h x and h s such that c(µ, v(x( )) Y) sup β 2 µ (v,q,y) V (v, q, Y) (H 1/2 (B) d ) V Daniele Boffi FD-DLM-IBM May 30, 2016 page 46

58 Proof of inf-sup Fix µ h Λ. The continuous inf-sup conditions implies that there exists ū H 1 0 (Ω) such that c(µh, ū(x)) β 1 µ Λ ū 1 ū h = Πū V h Clément interp. of ū, using inverse inequality we obtain c(µ h, ū h (X)) = c(µ h, ū(x)) + c(µ h, ū h (X) ū(x)) The result follows from c(µ sup h, v(x)) c(µ, ū h(x)) v V h v 1 ū h 1 if (h x /h s ) 1/2 β 1 /C β 1 µ Λ ū 1 C µ h 0,B hx 1/2 ū 1 ( ) ) 1/2 hx µ Λ ū 1 (β 1 C. ( β 1 C ( hx h s h s ) 1/2 ) ū 1 ū h 1 µ h Λ Daniele Boffi FD-DLM-IBM May 30, 2016 page 47

59 Error estimates Theorem The following error estimate holds true u u h H 1 0 (Ω) + p p h L 2 (Ω) + X X h H 1 (B) + λ λ h Λ ( C inf u v H 1 (v,q,y,µ) V h Q h S h S 0 (Ω) + p q L 2 (Ω) h ) + X Y H 1 (B) + λ µ Λ N.B.: the rate of convergence depends on the regularity of the solution Daniele Boffi FD-DLM-IBM May 30, 2016 page 48

60 The inflated balloon (thin solid) u(x, t) = 0 x Ω, t ]0, T[ { κ(1/r πr), x R p(x, t) = κπr, x > R Time convergence t ]0, T[ t X ex X h L 2 L 2 -rate u ex u h L 2 L 2 -rate e e e e e e e e Daniele Boffi FD-DLM-IBM May 30, 2016 page 49

61 The inflated balloon (thin solid) Spatial convergence h x p p h L 2 L 2 -rate u u h L 2 L 2 -rate 1/ / / / / / / Daniele Boffi FD-DLM-IBM May 30, 2016 page 50

62 Conclusions The use of a Lagrange multiplier can be successfully extended from interface to FSI problems Resulting semi-implicit scheme is unconditionally stable in time Analysis of stationary problem provides optimal error estimates Daniele Boffi FD-DLM-IBM May 30, 2016 page 51