Multigrid Methods for Saddle Point Problems

Multigrid Methods for Saddle Point Problems Susanne C. Brenner Department of Mathematics and Center for Computation & Technology Louisiana State University Advances in Mathematics of Finite Elements (In honor of the ninetieth birthday of Ivo Babuša)

Outline Problems Methods Analysis Numerics Conclusions

References B., Hengguang Li and Li-yeng Sung Multigrid methods for saddle point problems: Stoes and Lamé systems B., Du-Soon Oh and Li-yeng Sung Numer. Math. (2014) Multigrid methods for saddle point problems: Darcy systems Preprint

References Numer. Math. 20, t 79---t92 (t973) 9 by Springer-Verlag t973 The Finite Element Method with Lagrangian Multipliers* Ivo Babu~a Institut for Fluid Dynamic and Applied Mathematics, University of Maryland, College Par/U.S.A. Received January 26, t972 Summary. The Dirichlet problem for second order differential equations is chosen as a model problem to show how the finite element method may be implemented to avoid difficulty in fulfilling essential (stable) boundary' conditions. The implementation is based on the application of Lagrangian multiplier. The rate of convergence is proved. 1. Introduction The finite element method has become the most successfull approximation method in engineering. There is a variety of detailed approaches based on the finite element method. See e.g. [22 and 17] many others. The central idea of the finite element method is to use different variational principles together with a

References MATHEMATICS OF COMPUTATION, VOLUME 35, NUMBER 152 OCTOBER 1980, PAGES 1039-1062 Analysis of Mixed Methods Using Mesh Dependent Norms* By I. Babusa, J. Osborn and J. Pitaranta Abstract. This paper analyzes mixed methods for the biharmonic problem by means of new families of mesh dependent norms which are introduced and studied. More specifically, several mixed methods are shown to be stable with respect to these norms and, as a consequence, error estimates are obtained in a simple and direct manner. 1. Introduction. In [5] Brezzi studied Ritz-Galerin approximation of saddlepoint problems arising in connection with Lagrange multipliers. These problems have the form: Given fce V' and g E W', find (u, C) V x W satisfying (l.l) ~~a(u,v) + b(v, ;) (f, v) Vv G V, b(u, o)= (g, <0) Vfo E W,

Saddle Point Problems

Saddle Point Problems Find (u, p) V Q such that a(u, v) + b(v, p) = F (v) b(u, q) c(p, q) = G(q) v V q Q V and Q are Hilbert spaces. a(, ) is a bounded bilinear form on V V. b(, ) is a bounded bilinear form on V Q. c(, ) is a bounded bilinear form on Q Q. F is a bounded linear function on V. G is a bounded linear functional on Q. Goal Construct multigrid methods that converge uniformly in the energy norm V + Q for elliptic boundary value problems formulated as saddle point problems, without assuming full elliptic regularity.

Saddle Point Problems Find (u, p) V Q such that a(u, v) + b(v, p) = F (v) b(u, q) c(p, q) = G(q) v V q Q Most references in the literature (Verfürth, Wittum, Braess- Sarazin, Schöberl-Zulehner,...) only yield convergence in norms other than the energy norm and many require full elliptic regularity (convex domain).

Saddle Point Problem I Find (u, p) V Q such that a(u, v) + b(v, p) = F (v) v V b(u, q) = 0 q Q is a bounded polyhedral domain in R d. (d = 2, 3) V is a (closed) subspace of [H 1 ()] d. Q is a (closed) subspace of L 2 (). a(, ) is a symmetric bounded bilinear form on V V. b(, ) is a bounded bilinear form on V Q. F is a bounded linear functional on V.

Saddle Point Problem I Find (u, p) V Q such that a(u, v) + b(v, p) = F (v) v V b(u, q) = 0 q Q There exist positive constants γ and β such that Coercivity a(v, v) γ v 2 H 1 () v V Inf-Sup Condition inf sup q Q v V b(v, q) β v H 1 () q L2 ()

Saddle Point Problem I Find (u, p) V Q such that a(u, v) + b(v, p) = F (v) v V b(u, q) = 0 q Q Under these assumptions the saddle point problem is wellposed. Babuša 1973 Brezzi 1974

Saddle Point Problem I Find (u, p) V Q such that a(u, v) + b(v, p) = F (v) v V b(u, q) = 0 q Q Elliptic Regularity There exists α (0, 1] such that u H 1+α () + p H α () C F H 1+α ()

Saddle Point Problem I Stoes System (u = fluid velocity, p = pressure) Find (u, p) [H0 1()]d L 0 2 () such that u : v dx ( u)p dx = F (v) v [H0 1 ()] d ( u)q dx = 0 q L 0 2() is a bounded polyhedral domain in R d. (d = 2, 3) V = [H0 1()]d (no-slip) Q = L 0 2 () = {q L 2() : v dx = 0}

Saddle Point Problem I Stoes System (u = fluid velocity, p = pressure) Find (u, p) [H0 1()]d L 0 2 () such that u : v dx ( u)p dx = F (v) v [H0 1 ()] d ( u)q dx = 0 q L 0 2() a(w, v) = w : v dx b(v, q) = ( v) q dx

Saddle Point Problem I Stoes System (u = fluid velocity, p = pressure) Find (u, p) [H0 1()]d L 0 2 () such that u : v dx ( u)p dx = F (v) v [H0 1 ()] d ( u)q dx = 0 q L 0 2() Elliptic Regularity There exists α ( 1 2, 1] such that u H 1+α () + p H α () C F H 1+α ()

Saddle Point Problem I Discrete Problem Find (u h, p h ) V h Q h such that a(u h, v) + b(v, p h ) = F (v) v V h b(u h, q) = 0 q Q h V h Q h [H0 1()]d L 0 2 () is a stable pair for the Stoes system (say the P l -P l 1 (l 2) Taylor-Hood elements) i.e., b(v, q) inf sup q Q h v V h v H 1 () + q L2 () β d > 0 where the discrete inf-sup constant β d is independent of the mesh size h.

Saddle Point Problem I Discrete Problem Find (u h, p h ) V h Q h such that a(u h, v) + b(v, p h ) = F (v) v V h b(u h, q) = 0 q Q h V h Q h [H0 1()]d L 0 2 () is a stable pair for the Stoes system (say the P l -P l 1 (l 2) Taylor-Hood elements) i.e., b(v, q) inf sup q Q h v V h v H 1 () + q L2 () β d > 0 The coercivity of a(, ) and the discrete inf-sup condition imply that the discrete problem provides a stable approximation of the Stoes system.

Saddle Point Problem I Discrete Problem Find (u h, p h ) V h Q h such that a(u h, v) + b(v, p h ) = F (v) v V h b(u h, q) = 0 q Q h Compact Form B ( (u h, p h ), (v, q) ) = F (v) (v, q) V h Q h where B ( (u h, p h ), (v, q) ) = a(u h, v) + b(v, p h ) + b(u h, q)

Saddle Point Problem I Discrete Problem Find (u h, p h ) V h Q h such that a(u h, v) + b(v, p h ) = F (v) v V h b(u h, q) = 0 q Q h Compact Form B ( (u h, p h ), (v, q) ) = F (v) (v, q) V h Q h Stability Estimate sup (w,r) V h Q h ( ) B (v, q), (w, r) w H 1 () + r L2 () v H 1 () + q L2 () for all (v, q) V h Q h

Saddle Point Problem I Discrete Problem Find (u h, p h ) V h Q h such that a(u h, v) + b(v, p h ) = F (v) v V h b(u h, q) = 0 q Q h Compact Form B ( (u h, p h ), (v, q) ) = F (v) (v, q) V h Q h Quasi-Optimal Error Estimate u u h H 1 () + p p h L2 () C ( ) inf u v H v V 1 () + inf p q L2 () h q Q h

Saddle Point Problem I We can also allow a(, ) to be nonsymmetric in the saddle point problem a(u, v) + b(v, p) = F (v) v [H 1 0 ()] d b(u, q) = 0 q L 0 2() Oseen System a(u, v) = b(v, p) = u : v dx + ( v)p dx (w u) v dx w [W 1 ()] d H(div 0 ; ) is a wind function.

Saddle Point Problem I We can also consider a saddle point problem of a more general form a(u, v) + b(v, p) = F (v) b(u, q) c(p, q) = 0 v [H 1 0 ()] d q L 0 2() Lamé System a(u, v) = 2µ b(v, p) = c(p, q) = 1 λ ɛ(u) : ɛ(v) dx ( v)p dx pq dx

Saddle Point Problem II Find (u, p) V Q such that a(u, v) + b(v, p) = 0 v V b(u, q) = G(q) q Q is a bounded polyhedral domain in R d. (d = 2, 3) V is a (closed) subspace of [L 2 ()] d. Q is a (closed) subspace of H 1 (). a(, ) is a symmetric bounded bilinear form on V V. b(, ) is a bounded bilinear form on V Q. G is a bounded linear functional on Q.

Saddle Point Problem II Find (u, p) V Q such that a(u, v) + b(v, p) = 0 v V b(u, q) = G(q) q Q There exist positive constants γ and β such that Coercivity a(v, v) γ v 2 L 2 () v V Inf-Sup Condition inf sup q Q v V b(v, q) β v L2 () q H 1 ()

Saddle Point Problem II Find (u, p) V Q such that a(u, v) + b(v, p) = 0 v V b(u, q) = G(q) q Q Under these assumptions the saddle point problem is wellposed.

Saddle Point Problem II Find (u, p) V Q such that a(u, v) + b(v, p) = 0 v V b(u, q) = G(q) q Q Under these assumptions the saddle point problem is wellposed. Elliptic Regularity There exists α (0, 1] such that u H α () + p H 1+α () C G H 1+α ()

Saddle Point Problem II Darcy System (u = fluid velocity, p = pressure) Find (u, p) [L 2 ()] d H0 1 () such that K 1 u v dx v p dx = 0 v [L 2 ()] d u q dx = G(q) q H0 1 () is a bounded polyhedral domain in R d. (d = 2, 3) V = [L 2 ()] d Q = H0 1() K is a d d SPD matrix.

Saddle Point Problem II Darcy System (u = fluid velocity, p = pressure) Find (u, p) [L 2 ()] d H0 1 () such that K 1 u v dx v p dx = 0 v [L 2 ()] d u q dx = G(q) q H0 1 () a(w, v) = K 1 w v dx b(v, q) = v q dx

Saddle Point Problem II Darcy System (u = fluid velocity, p = pressure) Find (u, p) [L 2 ()] d H0 1 () such that K 1 u v dx v p dx = 0 v [L 2 ()] d u q dx = G(q) q H0 1 () Elliptic Regularity There exists α ( 1 2, 1] such that u H α () + p H 1+α () C G H 1+α ()

Saddle Point Problem II Darcy System (u = fluid velocity, p = pressure) Find (u, p) [L 2 ()] d H0 1 () such that K 1 u v dx v p dx = 0 v [L 2 ()] d u q dx = G(q) q H0 1 () The dual problem posed on H(div; ) L 2 () is the more attractive formulation of the Darcy system since it provides more information on the velocity u.

Saddle Point Problem II Darcy System (u = fluid velocity, p = pressure) Find (u, p) [L 2 ()] d H0 1 () such that K 1 u v dx v p dx = 0 v [L 2 ()] d u q dx = G(q) q H0 1 () Dual Formulation Find (u, p) H(div; ) L 2 () such that K 1 u v dx + ( v)p dx = 0 v H(div; ) ( u)q dx = G(q) q L 2 ()

Saddle Point Problem II Darcy System (u = fluid velocity, p = pressure) Find (u, p) [L 2 ()] d H0 1 () such that K 1 u v dx v p dx = 0 v [L 2 ()] d u q dx = G(q) q H0 1 () However we can treat conforming mixed finite element methods for the dual formulation posed on H(div; ) L 2 () as nonconforming mixed finite element methods for the formulation posed on [L 2 ()] d H 1 0 ().

Saddle Point Problem II Discrete Problem Find (u h, p h ) V h Q h such that K 1 u h v dx + ( v)p h dx = 0 v V h ( u h )q dx = G(q) q Q h V h = RT l ( H(div; )) is the Raviart-Thomas finite element space of order l 1. Q h is the space of (discontinuous) piecewise P l functions. V h Q h is a stable finite element pair for the Darcy system posed on H(div; ) L 2 () and the discrete problem is the standard Raviart-Thomas finite element method for this formulation of the Darcy system.

Saddle Point Problem II Discrete Problem Find (u h, p h ) V h Q h such that K 1 u h v dx + ( v)p h dx = 0 v V h ( u h )q dx = G(q) q Q h V h = RT l ( H(div; )) is the Raviart-Thomas finite element space of order l 1. Q h is the space of (discontinuous) piecewise P l functions. We will treat the discrete problem as a nonconforming method for the Darcy system posed on [L 2 ()] d H0 1 () by introducing mesh-dependent norms on V h and Q h.

Saddle Point Problem II Discrete Problem Find (u h, p h ) V h Q h such that K 1 u h v dx + ( v)p h dx = 0 v V h ( u h )q dx = G(q) q Q h Mesh-Dependent L 2 Norm for V h v 2 L 2 (;T h ) = T T h v 2 L 2 (T ) + σ S h h σ v n σ 2 L 2 (σ) S h is the set of the sides (edges/faces) of the elements in T h. n σ is a unit normal of σ and h σ is the diameter of σ.

Saddle Point Problem II Discrete Problem Find (u h, p h ) V h Q h such that K 1 u h v dx + ( v)p h dx = 0 v V h ( u h )q dx = G(q) q Q h Mesh-Dependent L 2 Norm for V h v 2 L 2 (;T h ) = T T h v 2 L 2 (T ) + σ S h h σ v n σ 2 L 2 (σ) This norm is well-defined for the velocity u = K p H α () (α > 1 2 ) in the continuous problem and it is also equivalent to L2 () on V h.

Saddle Point Problem II Discrete Problem Find (u h, p h ) V h Q h such that K 1 u h v dx + ( v)p h dx = 0 v V h ( u h )q dx = G(q) q Q h Mesh-Dependent H 1 Norm for Q h q 2 H 1 (;T h ) = T T h q 2 L 2 (T ) + 1 [q ] h σ 2 L 2 (σ) σ σ S h [q ] σ (a vector) is the jump of q across the side σ.

Saddle Point Problem II Discrete Problem Find (u h, p h ) V h Q h such that K 1 u h v dx + ( v)p h dx = 0 v V h ( u h )q dx = G(q) q Q h Mesh-Dependent H 1 Norm for Q h q 2 H 1 (;T h ) = T T h q 2 L 2 (T ) + 1 [q ] h σ 2 L 2 (σ) σ σ S h This is a standard DG norm for piecewise H 1 functions, which is well-defined on the pressure p in the continuous problem.

Saddle Point Problem II Discrete Problem Find (u h, p h ) V h Q h such that K 1 u h v dx + ( v)p h dx = 0 v V h ( u h )q dx = G(q) q Q h The bilinear form (w, v) is clearly bounded with respect to K 1 w v dx ( v L2 (;T h ) = v 2 L 2 (T ) + h σ v n σ 2 L 2 (σ) T T h σ S h on u + V h. ) 1 2

Saddle Point Problem II Discrete Problem Find (u h, p h ) V h Q h such that K 1 u h v dx + ( v)p h dx = 0 v V h ( u h )q dx = G(q) q Q h The bilinear form (w, v) is also coercive on V h : K 1 w v dx K 1 v v dx λ min (K 1 ) v 2 L 2 () v 2 L 2 (;T h ) v V h

Saddle Point Problem II It follows from the integration by parts formula ( v)q dx = ( (n v)q ds T T h = σ S h σ T v [q ] σ ds that the bilinear form (v, q) ( v)q dx T T h T ) v q dx T v q dx is bounded on ( u + V h ) ( p + Q h ) with respect to ( v L2 (;T h ) = v 2 L 2 (T ) + h σ v n σ 2 L 2 (σ) T T h σ S h ( q H 1 (;T h ) = q 2 1 L 2 (T ) + [q] h σ 2 L 2 (σ) σ T T h σ S h ) 1 2 ) 1 2

Saddle Point Problem II It follows from the integration by parts formula ( v)q dx = ( (n v)q ds T T h = σ S h σ T v [q ] σ ds that the bilinear form (v, q) ( v)q dx T T h satisfies an inf-sup condition ( v)q dx inf sup β d > 0 q Q h v V h v L2 (;T h ) q H 1 (;T h ) where β d is independent of h. T ) v q dx T v q dx

Saddle Point Problem II Discrete Problem Find (u h, p h ) V h Q h such that K 1 u h v dx + ( v)p h dx = 0 v V h ( u h )q dx = G(q) q Q h Compact Form B ( (u h, p h ), (v, q) ) = G(q) where B ( (u h, p h ), (v, q) ) = q Q h K 1 u h v dx + ( v)p h dx + ( u h )q dx

Saddle Point Problem II Discrete Problem Find (u h, p h ) V h Q h such that K 1 u h v dx + ( v)p h dx = 0 v V h ( u h )q dx = G(q) q Q h Stability Estimate ( ) B (v, q), (w, r) sup (w,r) V h Q h w L2 (;T h ) + r H 1 (;T h ) v L2 (;T h )+ q H 1 (;T h ) for all (v, q) V h Q h

Saddle Point Problem II Discrete Problem Find (u h, p h ) V h Q h such that K 1 u h v dx + ( v)p h dx = 0 v V h ( u h )q dx = G(q) q Q h Quasi-Optimal Error Estimate u u h L2 (;T h ) + p p h H 1 (;T h ) C ( ) inf u v L2 (;T v V h ) + inf p q H h q Q 1 (;T h ) h This is the reason why we do not use the lowest order Raviart- Thomas finite element pair, where Q h is the space of piecewise constant functions, because in that case inf q Qh p q H 1 (;T h ) does not go to 0 as h decreases to 0.

Saddle Point Problem II It follows from the definitions of the mesh-dependent norms ( v L2 (;T h ) = v 2 L 2 (T ) + h σ v n σ 2 L 2 (σ) T T h σ S h ( q H 1 (;T h ) = q 2 L 2 (T ) + 1 [q ] h σ 2 L 2 (σ) σ T T h σ S h and a Poincaré-Friedrichs inequality for piecewise H 1 functions that ) 1 2 ) 1 2 u u h L2 () + p p h L2 () C ( ) u u h L2 (;T h ) + p p h H 1 (;T h ) Hence the standard error estimate for the Raviart-Thomas finite element method follows from the error estimate in the meshdependent norms.

Saddle Point Problem II We can also consider a more general (nonsymmetric) discrete problem. Find (u h, p h ) V h Q h such that K 1 u h v dx + ( v)p h dx = 0 v V h ( u h )q dx (cp + b h p)q dx = G(q) q Q h where c W 1 (), b [W 1, ()] d H(div 0 ; ) and c 1 2 b 0 (Darcy system with convective and reactive terms)

Multigrid Methods

A Symmetric Positive Definite Problem Find u H0 1 () such that u v dx = fv dx v H 1 0 () is a polygonal domain in R 2. f belongs to L 2 (). Elliptic Regularity u H 1+α () C f H 1+α () for some α ( 1 2, 1]. (α = 1 for convex )

A Symmetric Positive Definite Problem Find u H0 1 () such that u v dx = fv dx v H 1 0 () Set-Up for Multigrid T 0 is a triangulation of. T 1, T 2,... are generated from T 0 by uniform subdivision. h is the mesh size. V 0 V 1 are nested P 1 finite element spaces.

A Symmetric Positive Definite Problem Find u H0 1 () such that u v dx = fv dx v H 1 0 () A Mesh-Dependent Inner Product [v, w] = h 2 v(p)w(p) p v V where the summation is over all the vertices of T. [v, v] v 2 L 2 () v V

A Symmetric Positive Definite Problem Find u H0 1 () such that u v dx = fv dx v H 1 0 () -th Level Problem ( ) A u = f where [A w, v] = a(v, w) = [f, v] = fv dx w v dx v, w V v V

Multigrid Algorithms Find u H0 1 () such that u v dx = fv dx v H 1 0 () -th Level Problem ( ) A u = f Apply pre-smoothing steps with initial guess v 0 to obtain an approximate solution v of ( ). Transfer the residual of v to a coarse grid and apply the multigrid algorithm on the coarse grid to find an approximate correction of v. Apply post-smoothing steps to the corrected approximate solution of ( ) to obtain the final output.

Multigrid Algorithms Two Ingredients Intergrid transfer operators Efficient smoother

Multigrid Algorithms Two Ingredients Intergrid transfer operators Efficient smoother Coarse-to-Fine Operator I 1 : V 1 V is the natural injection. (The finite element spaces are nested.)

Multigrid Algorithms Two Ingredients Intergrid transfer operators Efficient smoother Coarse-to-Fine Operator I 1 : V 1 V is the natural injection. Fine-to-Coarse Operator : V V 1 is the transpose of I 1 mesh-dependent inner product: [ I 1 v, w] 1 = [ v, I 1 w] v V, w V 1 I 1 with respect to the

Multigrid Algorithms Two Ingredients Intergrid transfer operators Efficient smoother Coarse-to-Fine Operator I 1 : V 1 V is the natural injection. Ritz Projection Operator P 1 : V V 1 is the transpose of I 1 variational bilinear form a(, ). with respect to the a ( P 1 v, w ) = a ( v, I 1 w) v V, w V 1

Multigrid Algorithms Smoothing Step for A u = f v new = v old + S (f A v old )

Multigrid Algorithms Smoothing Step for A u = f v new = v old + S (f A v old ) Richardson Relaxation S = γ Id where the damping factor γ = Ch 2 is chosen so that the spectral radius of S A is 1

Error Propagation for the Two-Grid Algorithm m pre-smoothing steps m post-smoothing steps Exact solve on the coarse grid E = R m (Id I 1 P 1 )R m R = Id S A = Id γ A measures the effect of one smoothing step. Id I 1 P 1 measures the effect of the exact coarse grid solve.

Error Propagation for the Two-Grid Algorithm E = R m (Id I 1 P 1 )R m Estimates for R m (smoothing property) and Id I 1 P 1 (approximation property) can be established in terms of a scale of mesh-dependent norms. Ban and Dupont 1981

Error Propagation for the Two-Grid Algorithm Mesh-Dependent Norms E = R m (Id I 1 P 1 )R m v s, = [ A s v, v] v V

Error Propagation for the Two-Grid Algorithm Mesh-Dependent Norms E = R m (Id I 1 P 1 )R m v s, = [ A s v, v] v V Connection to Sobolev Norms v 0, v L2 () v 1, = v H 1 () v V v V Interpolation between Hilbert Scales v s, v H s () v V for 0 s 1 (s 1 2 )

Error Propagation for the Two-Grid Algorithm Mesh-Dependent Norms E = R m (Id I 1 P 1 )R m v s, = [ A s v, v] v V Smoothing Property R m v s, (h m) s t v t, v V for 0 s t 2 R = Id γ A γ = Ch 2 and ρ(γ A ) 1

Error Propagation for the Two-Grid Algorithm Mesh-Dependent Norms E = R m (Id I 1 P 1 )R m v s, = [ A s v, v] v V Approximation Property (Id I 1 P 1 )v 1 α, (Id I 1 P 1 h α v H 1 () )v H 1 α () h α v 1, v V

Error Propagation for the Two-Grid Algorithm E = R m (Id I 1 P 1 )R m E v H 1 () = E v 1, definition of mesh-dependent norm

Error Propagation for the Two-Grid Algorithm E = R m (Id I 1 P 1 )R m E v H 1 () = E v 1, = R m (Id I 1 P 1 )(Id I 1 P 1 )R m v 1, Id I 1 P 1 = (Id I 1 P 1 ) 2

Error Propagation for the Two-Grid Algorithm E = R m (Id I 1 P 1 )R m E v H 1 () = E v 1, = R m (Id I 1 P 1 )(Id I 1 P 1 )R m v 1, (h m) α (Id I 1 P 1 )(Id I 1 P 1 )R m v 1 α, smoothing property R m v 1, (h m) α v 1 α,

Error Propagation for the Two-Grid Algorithm E = R m (Id I 1 P 1 )R m E v H 1 () = E v 1, = R m (Id I 1 P 1 )(Id I 1 P 1 )R m v 1, (h m) α (Id I 1 P 1 )(Id I 1 P 1 )R m v 1 α, (h m) α h α (Id I 1 P 1 )R m v 1, approximation property (Id I 1 P 1 )v 1 α, h α v 1,

Error Propagation for the Two-Grid Algorithm E = R m (Id I 1 P 1 )R m E v H 1 () = E v 1, = R m (Id I 1 P 1 )(Id I 1 P 1 )R m v 1, (h m) α (Id I 1 P 1 )(Id I 1 P 1 )R m v 1 α, (h m) α h α (Id I 1 P 1 )R m v 1, (h m) α h α hα Rm v 1+α, approximation property (Id I 1 P 1 )v 1, h α v 1+α,

Error Propagation for the Two-Grid Algorithm E = R m (Id I 1 P 1 )R m E v H 1 () = E v 1, = R m (Id I 1 P 1 )(Id I 1 P 1 )R m v 1, (h m) α (Id I 1 P 1 )(Id I 1 P 1 )R m v 1 α, (h m) α h α (Id I 1 P 1 )R m v 1, (h m) α h α hα Rm v 1+α, (h m) α h α hα (h m) α v 1, smoothing property R m v 1+α, (h m) α v 1,

Error Propagation for the Two-Grid Algorithm E = R m (Id I 1 P 1 )R m E v H 1 () = E v 1, = R m (Id I 1 P 1 )(Id I 1 P 1 )R m v 1, (h m) α (Id I 1 P 1 )(Id I 1 P 1 )R m v 1 α, (h m) α h α (Id I 1 P 1 )R m v 1, (h m) α h α hα Rm v 1+α, (h m) α h α hα (h m) α v 1, m α v H 1 () definition of mesh-dependent norm

Convergence Results It follows from the estimate E v H 1 () m α v H 1 () that the two-grid algorithm is a contraction with a contraction number bounded uniformly away from 1 provided the number of smoothing steps (independent of mesh levels) is sufficiently large. The same convergence result holds for the W -cycle algorithm by a perturbation argument. This result can be improved to uniform convergence with 1 smoothing step for both W -cycle and V -cycle algorithms by exploiting a strengthened Cauchy-Schwarz inequality. Bramble, Pascia, Wang, Xu, Zhang, Ziatanov,...

Summary The ey to the multigrid convergence analysis for second order symmetric positive definite problems is the existence of a scale of mesh dependent norms related to the operator S A, where S appears in the smoothing step v new = v old S (f A v old ) This scale of mesh dependent norms should be equivalent to the scale of Sobolev norms between L 2 () and H 1 (). Then we can establish the smoothing property for the operator R = Id S A and the approximation property for the operator Id I 1 P 1

Summary The ey to the multigrid convergence analysis for second order symmetric positive definite problems is the existence of a scale of mesh dependent norms related to the operator S A, where S appears in the smoothing step v new = v old S (f A v old ) This scale of mesh dependent norms should be equivalent to the scale of Sobolev norms between L 2 () and H 1 (). Note that we can prove convergence of the multigrid methods in the energy norm (H 1 norm) because it is equivalent to the (1, ) mesh-dependent norm.

Multigrid Methods for Saddle Point Problems T 0 is a triangulation of. T 1, T 2,... are generated from T 0 by uniform subdivision. h is the mesh size. V 0 Q 0 V 1 Q 1 are nested finite element spaces. (Taylor-Hood for Stoes/Raviart-Thomas for Darcy)

Multigrid Methods for Saddle Point Problems -th Level Discrete Problem Find (u, p ) V Q such that B ( (u, p ), (v, q) ) = F (v) + G(q) (v, q) V Q The bilinear form B(, ) satisfies the stability estimate ( ) B (v, q), (w, r) sup v (w,r) V Q w V + r V + q Q Q for all (v, q) V Q.

Multigrid Methods for Saddle Point Problems -th Level Discrete Problem Find (u, p ) V Q such that B ( (u, p ), (v, q) ) = F (v) + G(q) (v, q) V Q System Operator B We can represent the bilinear form B(, ) on V Q by the operator B : V Q V Q defined by [ B (v, q), (w, r) ] = B( (v, q), (w, r) ) (v, q), (w, r) V Q where [, ] is a mesh-dependent inner product.

Multigrid Methods for Saddle Point Problems The mesh-dependent inner product [, ] on V Q satisfies the following norm equivalence. Stoes [ (v, q), (v, q) ] v 2 L 2 () + h2 q 2 L 2 () For the Stoes problem, the elliptic regularity result holds only for the v component. Therefore the Aubin-Nitsche duality argument can only be applied to the approximation property involving v. By including h 2 with q 2 L 2 (), the approximation property involving q becomes trivial.

Multigrid Methods for Saddle Point Problems The mesh-dependent inner product [, ] on V Q satisfies the following norm equivalence. Stoes [ ] (v, q), (v, q) v 2 L 2 () + h2 q 2 L 2 () Darcy [ ] (v, q), (v, q) h2 v 2 L 2 () + q 2 L 2 () For the Darcy problem, the elliptic regularity result holds for the q component and hence we include h 2 with v 2 L 2 ().

Multigrid Methods for Saddle Point Problems The mesh-dependent inner product [, ] on V Q satisfies the following norm equivalence. Stoes [ ] (v, q), (v, q) v 2 L 2 () + h2 q 2 L 2 () Darcy [ ] (v, q), (v, q) h2 v 2 L 2 () + q 2 L 2 () We can define these mesh-dependent inner products by mass lumping so that the matrices representing them with respect to the standard DOFs are diagonal matrices.