Introduction to Domain Decomposition Methods in the numerical approximation of PDEs. Luca Gerardo-Giorda

Introduction to Domain Decomposition Methods in the numerical approximation of PDEs Luca Gerardo-Giorda

Plan 0 Motivation Non-overlapping domain decomposition methods Multi-domain formulation Variational formulation of the multi-domain problem Iterative methods Non-overlapping methods at the finite dimensional level Finite dimensional formulation of the single domain problem Finite dimensional multi-domain formulation Iterative substructuring methods at the finite dimensional level 3 Overlapping Schwarz Methods Classical Schwarz methods Convergence of the Classical Schwarz method 4 Optimal and Optimized Schwarz Methods Optimal Schwarz Method Optimized Schwarz Methods 5 Applications of Optimized Schwarz Methods Maxwell equations Fluid-Structure Interaction problems in haemodynamics Heterogeneous models coupling in electrocardiology

3. Overlapping Schwarz Methods

When everything started (869) Question: Existence of harmonic function on non-lipschitz domains? Dirichlet principle. The solution of Laplace equation u = 0 on a bounded domain Ω with Dirichlet boundary conditions u = g on Ω is the infimum of the Dirichlet integral Ω v dx over all the functions v satisfying the boundary condition v = g on Ω. K. H. A. Schwarz (843-9) In 869 Schwarz introduces a method to prove that the infimum is actually attained.

Classical Alternating Schwarz Method The method is based on decomposing a general domain Ω into two overlapping domains with Lipschitz boundary Ω = Ω Ω.

Classical Alternating Schwarz Method The method is based on decomposing a general domain Ω into two overlapping domains with Lipschitz boundary Ω = Ω Ω. u = 0 in Ω u = g on Ω Ω u = u0 on Γ solve on the disk

Classical Alternating Schwarz Method The method is based on decomposing a general domain Ω into two overlapping domains with Lipschitz boundary Ω = Ω Ω. u = 0 in Ω u = g on Ω Ω u = u on Γ solve on the rectangle

Classical Alternating Schwarz Method The method is based on decomposing a general domain Ω into two overlapping domains with Lipschitz boundary Ω = Ω Ω. u n = 0 in Ω u n = 0 in Ω u n = g on Ω Ω u n = g on Ω Ω u n = un on Γ solve on the disk u n = un on Γ solve on the rectangle In 869 Schwarz proves convergence of the iterative procedure using the maximum principle.

Classical Schwarz Method Model problem u + η u = f on Ω = R lim x u(x) < + Domain decomposition: Ω =(, L) R Ω =(0, ) R Classical Schwarz Algorithm u n + η un = f in Ω u n + η un = f in Ω lim x u n (x) < lim x u n (x) < u n (L, y) = un (L, y) y R u n (0, y) = un (0, y) y R

Classical Schwarz Method: Convergence Fourier transform along the interface û(x, k) = e iky u(x, y) dy Error Equation ê n = û û n ê n = û û n (η + k xx )ê n = 0 x < L, k R (η + k xx )ê n = 0 x > 0, k R ê n (L, k) =ên (L, k) k R e n (0, k) =en (0, k) k R The solution is given by k ê n (x, k) =A (k) e +η (x L) + B (k) e k +η (x L) k ê n (x, k) =A (k) e +η x + B (k) e k +η x

Classical Schwarz Method: Convergence From boundedness assumption: ê n (x, k) =ên (L, k)e k +η (x L) ê n (x, k) =ên (0, k)e k +η x By induction: where ê n (0, k) = ρ n cs ê 0 (0, k) ê n (L, k) = ρ n cs ê 0 (L, k) ρ cs = ρ cs (k, L, η) =e k +η L < k R For the Classical Schwarz Alternating Method overlapping is mandatory for convergence

Classical Schwarz Method 0.9 0.8 0.7 0.6! (k) 0.5 0.4 0.3 0. 0. 0 0 50 00 50 00 50 300 k Fast convergence for high frequencies, slow convergence for low frequencies

Classical Schwarz Method

Nonoverlapping Schwarz Method Pierre-Louis Lions (990) On the Schwarz Alternating Method III: a variant for nonoverlapping subdomains Robin interface conditions n u n + α u n = n u n + α u n n u n + α u n = n u n + α u n ensure convergence also without overlap First of all, it is possible to replace the constants in the Robin conditions by two proportional functions on the interface, or even by local or nonlocal operators.

4. Optimal and Optimized Schwarz Methods

Modified Schwarz Method u n + η un = f in Ω ( x + S ) u n (L, y) =( x + S ) u n (L, y) y R u n + η un = f in Ω ( x + S ) u n (0, y) =( x + S ) u n (0, y) y R A Fourier Transform provides the error equation (η + k xx )ê n (x, k) =0 x < L, k R ( x + σ (k)) ê n (L, k) =( x + σ (k)) ê n (L, k) k R (η + k xx )ê n (x, k) =0 x < L, k R ( x + σ (k)) ê n (0, k) =( x + σ (k)) ê n (0, k) k R

Modified Schwarz Method The solutions are given by ê n (x, k) =ên (L, k) σ (k) k +η e σ (k)+ k +η ê n (x, k) =ên (0, k) σ (k)+ σ (k) k +η (x L) k +η k +η e k +η x By induction: ê n (0, k) = ρ n os ê 0 (0, k) ê n (L, k) = ρ n os ê 0 (L, k) where ρ os = ρ os (k, L, η, σ, σ )= σ (k) σ (k)+ k +η k +η σ (k)+ σ (k) k +η k +η e k +η L ρ os (k, L, η, σ, σ )= σ (k) σ (k)+ k +η k +η σ (k)+ σ (k) k +η k +η ρ cs(k, L, η)

Optimal Schwarz Method ρ os (k, L, η, σ, σ )= σ (k) σ (k)+ k +η k +η σ (k)+ σ (k) k +η k +η e k +η L σ (k) = k + η σ (k) = k + η = ρ os (k, L, η, σ, σ ) 0 Convergence in two iterations, independently from initial guess overlap size L (allows nonoverlapping decompositions) problem coefficient η Optimal but not viable k + η is a nonlocal operator û n x = k + η û n û n x = k + η û n k + η Steklov-Poincaré operator

Optimized Schwarz Method We need viable approximation of the exact operators: σ (k) = k + η σ (k) = k + η Low frequency approximations (based on Taylor expansion) σ (k) = η + η k + O(k 4 ) σ (k) = η η k + O(k 4 ) Zero-th order (Robin condition) ρ T 0 (k, L, η) = k + η η k + η + η ρ cs (k, L, η) Order (differential operator on the interface) k + η η ρ T (k, L, η) = η k k + η + η + η k ρ cs (k, L, η)

Optimized Schwarz Method (L = 0) We need viable approximation of the exact operators: σ (k) = k + η σ (k) = k + η Low frequency approximations (based on Taylor expansion) σ (k) = η + η k + O(k 4 ) σ (k) = η η k + O(k 4 ) Zero-th order (Robin condition) ρ T 0 (k, L, η) = k + η η k + η + η Order (differential operator on the interface) k + η η ρ T (k, L, η) = η k k + η + η + η k

Optimized Schwarz Method 0.9 Classical Schwarz Taylor order 0 Taylor order 0.8 0.7 0.6! (k) 0.5 0.4 0.3 0. 0. 0 0 50 00 50 00 50 300 k

Optimized Schwarz Method (L = 0) 0.8 0.6! (k) 0.4 0. Classical Schwarz Taylor order 0 Taylor order 0 0 50 00 50 00 50 300 k

Optimized Schwarz Method Uniform approximation σ app (k) =p + q k σ app (k) = p q k k ρ = + η p q k k + η p q k ρ cs (k, L, η) k + η + p + q k k + η + p + q k Optimization as solution of a min-max problem Zero-th order conditions (q = q = 0) min p,p 0 max k min k kmax k + η p k + η p e k +η L k + η + p k + η + p Second order conditions min p j,q j 0 max k min k kmax k + η p q k k + η + p + q k k + η p q k k + η + p + q k e k +η L

Optimized Schwarz Method (p = p, q = q = 0) Theorem (M. J. Gander) For L > 0 and k max =, the solution p of the min-max problem is given by the unique root of the equation ρ Opt0 (k min, L, η, p )=ρ Opt0 ( k(l, η, p ), L, η, p ) k(l, η, p) = L(p + L(p η)) L For L = 0 and k max finite, the optimal parameter p is given by p = /4 k min + η kmax + η

Optimized Schwarz Method (L = 0) 0.8 0.6! (k) 0.4 0. Classical Schwarz Taylor order 0 Taylor order Optimized order 0 0 0 50 00 50 00 50 300 k Equioscillation: ρ Opt0 (k min, L, η, p )=ρ Opt0 (k max, L, η, p )

Optimized Schwarz Method (p = p, q = q ) From: M.J. Gander, Optimized Schwarz Methods, SIAM J. Num. Anal. (006)

Optimized Schwarz Method

Algebraic formulation of the Optimized Schwarz Method The Dirichlet model problem we are considering is η u ν u = f in Ω u Ω = 0 The sequential version of the OSM reads as follows Given λ 0 = nt ν u0 + α u 0 on Γ Solve for p 0. η u p+ ν u p+ = f in Ω n T ν up+ + α u p+ = λ p on Γ. λ p+ = n T ν up+ + α u p+ 3. η u p+ ν u p+ = f in Ω n T ν up+ + α u p+ = λ p+ on Γ 4. λ p+ = n T ν up+ + α u p+ Remark The normal derivatives in the interface conditions emerge naturally in the variational formulation The traces in the interface conditions have to be included by adding a term in the variational formulation

Algebraic formulation of the Optimized Schwarz Method Let us consider a non-overlapping decomposition. The bilinear forms within Ω j (j =, ) a j (u, v) = η Ω j uv M j Mass matrix + Ω j ν u v S j Stiffness matrix (n T ν u) v + Ω j Γ (n T ν u) v yield subdomain matrices B j = η M j + S j that already include the normal derivative on the interface Γ.

Algebraic formulation of the Optimized Schwarz Method Let us consider a non-overlapping decomposition. The bilinear forms within Ω j (j =, ) a j (u, v) = η Ω j uv M j Mass matrix + Ω j ν u v S j Stiffness matrix yield subdomain matrices B j = η M j + S j that already include the normal derivative on the interface Γ. To include the trace component we add a mass matrix defined on the interfaces nodes. [M Γ ] ij = Γ φ (Γ) j Γ φ(γ) i Γ

Algebraic formulation of the Optimized Schwarz Method We also need the restriction matrices, from the nodes in Ω j to the interface Γ. Let N j and N Γ be the number of internal and interface degrees of freedom for Ω j (j =, ), respectively. After a proper reordering of the unknowns, they are the Boolean operators R jγ =[0 Id ]= 0 0....... 0 0 0 N Γ N j... N Γ N Γ Their transposes are the zero-extension matrices from the interface Γ to Ω j. For u = uj u Γ : R jγ R T jγ u Γ = u Γ R T jγ R jγ uj u Γ = 0 uγ We can now write the subdomain matrices associated to Robin interface conditions, as A = B + α R T Γ M Γ R Γ A = B + α R T Γ M Γ R Γ

Algebraic formulation of the Optimized Schwarz Method We are now in the position of stating the fully discrete formulation of the Optimized Schwarz Method in sequential iterative form. Given λ 0 Solve for p 0 A u p+ = f + R T Γ λp λ p+ = R Γ (f B u p+ )+α M Γ R Γ u p+ 3 A u p+ = f + R T Γ λp+ 4 λ p+ = R Γ (f B u p+ )+α M Γ R Γ u p+

Optimized Schwarz Method as a preconditioner The parallel formulation of the Optimized Schwarz Method reads: A u p+ = f + R T Γ λp A u p+ = f + R T Γ λp λ p+ = R Γ (f B u p+ )+α M Γ R Γ u p+ λ p+ = R Γ (f B u p+ )+α M Γ R Γ u p+ Recalling the definition of A and A : A = B + α R T Γ M Γ R Γ A = B + α R T Γ M Γ R Γ, from the linear systems above we have u p+ = A f + A RT Γ λp B u p+ = f + R T Γ λp α R T Γ M Γ R Γ u p+ u p+ = A f + A RT Γ λp B u p+ = f + R T Γ λp α R T Γ M Γ R Γ u p+ We can now replace those quantities in the definitions of the interface variables.

Optimized Schwarz Method as a preconditioner u p+ = A f + A RT Γ λp B u p+ = f + R T Γ λp α R T Γ M Γ R Γ u p+ λ p+ = R Γ (f B u p+ )+α M Γ R Γ u p+ B u p+ = f + R T Γ λp α R T Γ M Γ R Γ u p+ = R Γ (f f R T Γ λp + α R T Γ M Γ R Γ u p+ )+α M Γ R Γ u p+ = R Γ R T Γ Id Γ p λ + α R Γ R T Γ Id Γ p+ M Γ R Γu + α M Γ R Γ u p+ = λ p +(α + α ) M Γ R Γ u p+ u p+ = A (f + R T Γ λp ) = λ p +(α + α ) M Γ R Γ A (f + R T Γ λp ) = Id +(α + α ) M Γ R Γ A RT Γ λ p +(α + α ) M Γ R Γ A f

Optimized Schwarz Method as a preconditioner Solving for λ p+ and λ p+ amounts to solve: λ p+ = λ p+ = Id +(α + α ) M Γ R Γ A RT Γ λ p +(α + α ) M Γ R Γ A f Id +(α + α ) M Γ R Γ A RT Γ λ p +(α + α ) M Γ R Γ A f Such algorithm is actually a fixed point iteration procedure to solve I I (α + α ) M Γ Σ I (α + α ) M Γ Σ I λ λ =(α + α ) R,Γ A f R,Γ A f where Σ = R,Γ A RT,Γ Σ = R,Γ A RT,Γ.

Classical Schwarz Method as a preconditioner Despite not being convergent without overlap in the iterative form, the Classical Schwarz can be used as a preconditioner for a Krylov acceleration procedure to solve the interface equation. In this case, the matrices associated to the Dirichlet Aii A A j = iγ problems on the interface of Ω j (j =, ) are given by 0 Id The parallel version of the Classical Schwarz Method reads as follows.. A u p+ = f + R T,Γ λp A u p+ = f + R T,Γ λp. λ p+ = R,Γ u p+ λ p+ = R,Γ u p+ Usiing. we solve for λ p+ and λ p+ : λ p+ = R,Γ A f + A RT,Γ λp λ p+ = R,Γ A f + A RT,Γ λp Fixed point iteration procedure to solve I R,Γ A R T,Γ R,Γ A R T,Γ I λ λ = R,Γ A f R,Γ A f

Numerical comparison 0 Classical Schwarz Classical Schwarz with Krylov Multigrid 0 Optimized Schwarz Optimized Schwarz with Krylov 0 0 0 0 0 3 0 4 0 5 0 6 0 4 6 8 0 4 6 8 0

Optimized Schwarz Method The optimized interface condition is given by n u n + Sopt un = n un + S opt un n u n + Sopt un = n un + S opt un where S opt j is a viable approximation of the Steklov-Poincaré operator in Ω j. PDE Optimized interface operator u + ηu = f () n u +[p (τ )(τ )] u ν u + b u = f () n u + n b p ν u u + ω u = f (3) n u + [(p + iq) (α + iβ)(τ )(τ )] u curl curl u ω u = f (4) (curl u) n + p+iq curl Γ curl Γ ω I (n u n) () : M.J. Gander, SIAMJ.NUM. ANAL. (006) (3) : M.J. Gander, F. Magoulès, F. Nataf, SIAMJ.SCI. COMPUT. (00) () : O. Dubois, DDMIN SCIENCE AND ENGINEERING, (007) (4) : A. Alonso-Rodrìguez, L.GG, SIAMJ.SCI. COMPUT. (006)

How to choose parameters in general

General settings of an Optimized Schwarz Method When the operator is elliptic, the algorithm is based on a combination of Fluxes and Traces PDE Optimized interface operator u + ηu = f n u +[p (τ )(τ )] u ν u + b u = f n u + n b p ν u u + ω u = f n u + [(p + iq) (α + iβ)(τ )(τ )] u curl curl u ω u = f (curl u) n + p+iq curl Γ curl Γ ω I (n u n) When the operator is hyperbolic, the algorithm is based on a combination of incoming characteristics and outcoming characteristics

5. Applications of Optimized Schwarz Methods

Optimized Schwarz Method for the full Maxwell system Time-Harmonic Maxwell Equations: iωεe + curl H σe = J iωµh + curl E = 0 Combination of incoming and outcoming characteristic variables. Solve for n =,,... Dirichlet coupling conditions Hyperbolic problems Characteristic variables Maxwell Equation Impedance conditions B nj (E, H) :=n j ε µ E + n j (H n j ). iωεe,n + curl H,n σe,n = J in Ω iωµh,n + curl E,n = 0 in Ω (Bn + S Bn )(E,n, H,n ) = (Bn + S Bn )(E,n, H,n ) on Γ iωεe,n + curl H,n σe,n = J in Ω iωµh + curl E = 0 in Ω (Bn + S Bn )(E,n, H,n ) = (Bn + S Bn )(E,n, H,n ) on Γ, S, S : Tangential operators Possibly pseudodifferential Based on Transparent b.c.

Transparent Boundary Conditions Fourier Analysis: Ω = R 3 Ω =(, L) R, Ω =(0, ) R J, compactly supported in Ω, lim r r H, n E, = 0 ω = ω µε, k = k y + k z (tangential frequency) Theorem (V. Dolean, M.J. Gander, L.GG, SIAM J. Sci. Comp. 009) Let the operators S l,l=,, have the Fourier symbol k F(S l )= ( y k z k y k z k ω + i ω) k y k z kz ky. Then, the solution of Maxwell s equations in Ω, with the boundary conditions (Bn + S Bn )(E (0, y, z), H (0, y, z)) = 0 (y, z) R (Bn + S Bn )(E (L, y, z), H (L, y, z)) = 0 (y, z) R coincides with the restriction on Ω, of the solution of Maxwell s equations in R 3.

Optimized Schwarz Method for Maxwell equations Approximate the nonlocal part of the symbols. k F(S l ):=γ y k z k y k z l k y k z kz ky γ l C(k z, k y ) # γ γ ρ # (k) 0 P(k) iω εµ P(k)+iω εµ Λ(k, L) s iω εµ k s+iω εµ P(k) s P(k)+s Λ(k, L) 3 k ω εµ+iω εµs P(k) iω εµ P(k)+iω εµ ρ ( k ) s 4 iω εµ k s +iω εµ s iω εµ k s +iω εµ P(k) s P(k)+s P(k) s P(k)+s Λ(k, L) 5 k ω εµ+iω εµs k ω εµ+iω εµs P(k) iω εµ P(k)+iω εµ ρ 4 ( k ) P(k) = k ω εµ Λ(k, L) =e P(k)L

Optimized Schwarz Method for Maxwell equations: asymptotics with overlap, L = h without overlap, L = 0 Case ρ parameters ρ parameters k + ω h none none C 6 ω h 3 p = C ω h 3 3 C CC 4 4 ω h p = ω C h 3 (k + ω ) 4 5 C 0 ω h 5 5 6 h 5 (k+ ω ) 0 h 5 3 p = (k + ω ) h 3 p = p = C 5 ω 7 5 h C 5 ω 6 5 h 5 3 5 3 p = (k + ω ) 5 7 5 h 5 p = (k + ω ) 5 6 3 5 h 5,, (k + ω ) 4 C 8 ω C 4 C(k h p = + ω ) C h h (k + ω ) C 4 4 8 h 4 3 p = C 8 ω C 4 h 4 3 p = C 8 ω C 4 3 h 4 p = (k + ω ) h 4 4, 3 8 C 4 3 p = (k + ω ) 8 C 4 3 h 4,

Numerical results Asymptotics!" # -,./0&*&%*+3'(4'+'5*(&$'*3'(6)00%+4)64*(%&$/7&%/'!$)(/*+%),' ),'! ),'# ),'8 ),'9 ),': ;<$!! = ;<$!!>8 = ;<$!!>8 = ;<$!!>: = ;<$!!>: =!" # -,./0&*&%*+3'(4'+'5*(&$'+*+!*3'(6)00%+4)64*(%&$/7&%/'!$)(/*+%),' ),'# ),'8 ),'9 ),': ;<$!!=# > ;<$!!=# > ;<$!!=9 > ;<$!!=9 %&'()&%*+, %&'()&%*+,!"!!"!# $!"!# $ 3d tests: Ω =(0, ) 3, ε = µ =, σ = 0 Ω =(0, β) (0, ) Ω =(α, ) (0, ) (0 < α β < ) Uniform mesh, FV scheme (INRIA, Sophia Antipolis) it S (it GS ) with overlap, L = h without overlap, L = 0 h /8 /6 /30 /8 /6 /30 Case 9(3) 9(7) 46() (93) (40) (0) Case 4() 9(4) 3(6) 48(9) 69(36) 98(48) Case 3 6() 8(4) (6) 65(35) 80(4) 66(55) Case 4 5(3) 9(5) (7) 38(8) 60(33) 04(39) Case 5 6(3) 8(4) (6) 70(36) 80(4) 76(55) it S : iterative version it GS : accelerated (GMRES) version

Chicken heating in a Whirlpool TC4 Microwave Ω =[0, 0.3] [0, 0.36] [0, 0.0]m Magnetron size 0.08 0.04m, metallic b.c. ε C = 4.43 0 farad σ C = 3 0 m siemens m farad m ε A = 8.85 0 σ A = 0 siemens m µ C = µ A = 4π 0 ω = π.45 GHz 7 henry m subdomains h = 0.005m Electrical Field Magnetic Field

Fluid Structure Interaction Problem Fluid-structure problem. Find the fluid velocity u, pressure p and the structure displacement η such that D A u ρ f + ρ f ((u w) )u T f = f f in Ω t (0, T ), Dt f u = 0 in Ω t (0, T ), f η ρ s t T s = f s in Ω 0 s (0, T ), () u = η on Σ t (0, T ), t T f n T s n = 0 on Σ t (0, T ), Geometry problem. Given η Σ t, find a map A : Ω 0 f Ω t f A t (x 0 )=x 0 + Ext( η Σ 0 ), such that Ω t f = A t (Ω 0 f ).

Time discretization and Fluid Structure interface treatment Given Ω f, u, w, u n, η n and η n, find u n+, p n+, and η n+ such that ρ f δ t u n+ + ρ f ((u w ) )u n+ T n+ f = f n+ f in Ω f, u n+ = 0 in Ω f, ρ s δ tt η n+ n+ n+ T s = f s in Ω 0 s, u n+ = δ t η n+ on Σ, T n+ f n = T n+ s n on Σ, with suitable boundary conditions for the artificial sections Γ t j, Γ j,s. Here: δ t w n+ = w n+ w n t δ tt w n+ = w n+ w n + w n t

Partitioned procedures Dirichlet-Neumann partitioned algorithms solve Fluid with the Structure velocity as Dirichlet b.c. solve Structure with the Fluid normal stress as Neumann b.c. Slow convergence and strong relaxation needed in the presence of large added mass effect. Robin-Robin partitioned algorithms At each iteration m: find u m, p m and η m such that Fluid Structure ρ f t u m + ρ f ((u w ) )u m T f,m = f f + ρ f t un in Ω f, u m = 0 in Ω f, η m η n α f u m T f,m n = α f T s,m n, on Σ, t ρ s t η m T s,m = f s + ρ s t ( ηn η n ) in Ω 0 s, α s t η m + T s,mn = α s t ηn + α s u m + T f,m n, on Σ The convergence is driven by the choice of α f and α s

. Potential Flow - Generalized String (P/GS) ρ f δ t u + p = 0 in Ω f = {x < 0} R, u = 0 in Ω f, u n = δ t η on Σ = {x = 0} R, p = ρ s H s δ tt η + βh s η GH s yy η, on Σ Robin Interface conditions: linear combinations with (α f, ), (α s, ) αf α f t δ t u x p = t ρ sh s t βh s η + GH s yy η + F (ux n, ηn, η n ), ρs H s t + βh s + α s η GH s yy η = α s t δ t u x + p + F (u n, η n, η n ), t here: u x = u n

. Potential Flow - Generalized String (P/GS) Since on Σ x p = ρ f δ t u x we can handle the interface condition in terms of the sole pressure p By applying the divergence operator and the Fourier Transform in the y direction, we get xx p m+ + k p m+ = 0 in (, 0) p m+ + α f t ρ f x p m+ = ρs H s t ρs H s t + βh s + GH s k + α s t + βh s + GH s k α f η m, for x = 0 t η m = p m α s t ρ f x p m for x = 0. We have p m+ (x, k) =A m+ (k) e kx where A m+ (k) is determined by the interface conditions x p m+ = A m+ (k) ke kx

. Potential Flow - Generalized String (P/GS) Then ρf + α f tk A m+ (k) = ρs H s t ρ f ρs H s t + βh s + GH s k + α s t + βh s + GH s k α f η m, for x = 0, t η m = ρ f α s tk ρ f A m (k), for x = 0. We thus have A m+ (k) = ρ(k) A m (k) The reduction factor of the iterative algorithm is independent of the iteration and is given by ρ P/GS α (k) = f F GS (k) α s + F GS (k) α s F P (k) α f + F P (k) Here: F GS (k) = ρ sh s t The algorithm converges provided ρs H s 0 α f < + βh s t + GH s tkmax t + βh s t + GH s tk F P (k) = ρ f t k 0 α s < ρ f α f + α s > 0 t k max

. Potential Flow - Generalized String (P/GS): parameter optimization We seek for α f and α s such that ρ P/GS (k) is minimal, in A {α(k) =α M + α K k, α M, α K 0} corresponding to a generalized Robin boundary condition: α M u α K Σ u + T f n = f (η) We have: ρ P/GS (k) = α f F GS (k) α s + F GS (k) α s F P (k) α f + F P (k) = 0 for α opt f Thus = F GS (k) = ρ sh s t + βh s t + GH s t k α GS f,m = ρ sh s t + βh s t = α heur f α GS f,k = GH s t

. Potential Flow - Generalized String (P/GS): parameter optimization α opt s = F P (k) = ρ f t k = integral boundary condition. Find α s through a minimization problem α opt s (k) = argmin max sup α s (k) A k [0,k max ] α f A α s F P (k) α f + F P (k) = argmin α s (k) A max k [0,k max ] α s t k ρ f If α s,k = 0 then ρ P/GS is unbounded for k = α P s = α s,m We have max k [0,k max ] α s t k ρ f = if α s,m ρ f t k max > otherwise = α P s = ρ f t k max

. Stokes - Generalized String (S/GS) ρ f δ t u u + p = 0 in Ω f = {x < 0} R, u = 0 in Ω f, u n = δ t η on Σ = {x = 0} R, p = ρ s H s δ tt η + βh s η GH s yy η, on Σ Reduction factor: ρ S/GS (k) = α f F GS (k) α s + F GS (k) α s F S (k) α f + F S (k) where F S (k) = µγ f (k)(k + γ f (k)) k γ f (k) = ρf µ t + k α opt = F f GS (k) as before = α GS f = ρ sh s t + βh s t + GH s t k α opt s from minimization α S s = µ t ρf t k + µ t (k ) k + ρ f + µ t (k ).

3. Stokes - Linear Incompressible Elasticity (S/LIE) ρ f δ t u u + p = 0 in Ω f = {x < 0} R, u = 0 in Ω f, ρ s δ tt η λη + βη + χ = 0 in Ω s =(0, H s ) R, η = 0 in Ω s Reduction factor: ρ S/LIE (k) = α f F LIE (k) α s + F LIE (k) α s F S (k) α f + F S (k) where: ε(k) = kγ s H k +γs s + O(Hs ), F LIE (k) = ε(k)+ t δ(k) = (ε(k)+) γ sh s + O(H δ(k) λγ s γ s H s 6 s ) γ s (k) = ρs λ t + β λ + k. For thin structures there is no significant improvement with respect to the GS model.

Numerical results I: rectangular domain Ω f 0 = 6 cm, Ωs 0 = 6 H s cm, ρ f = g/cm 3 ρ s =. g/cm 3 µ = 0.035 poise E =.3 0 6 dyne/cm 3 H s = 0. cm t = 0 3 s ν = 0.3, R = 0.5, β = E ( ν )R P(t) =0 4 dyne/cm for t 0.005 s Average number of iterations and (in brackets) relative CPU time α f = α s = 0 0 α P s α S s Ref. 5.87 (.00) 5.37 (0.9) 4.6 (0.79) 4.50 (0.77) ρ s = 0. 6.00 (.00) 5.6 (0.94) 4.87 (0.8) 4.87 (0.8) ρ s = 0.0 6.00 (.00) 5.75 (0.96) 5.00 (0.83) 4.87 (0.8) H s = 0.05 6.75 (.00) 5.87 (0.87) 5.00 (0.74) 5.00 (0.74) H s = 0.05 0.75 (.00) 7.87 (0.73) 7.6 (0.7) 5.75 (0.53) t = 5 0 4 7.00 (.00) 5.69 (0.8) 4.94 (0.7) 4.87 (0.70) t =.5 0 4 6.94 (.00) 5.56 (0.80) 4.8 (0.69) 4.4 (0.64) E = 6.5 0 5 6.50 (.00) 5.6 (0.86) 4.87 (0.75) 4.87 (0.75) E =.3 0 5 7.50 (.00) 5.50 (0.73) 4.37 (0.58) 5.00 (0.67) ν = 0.49 5.6 (.00) 4.87 (0.87) 4.00 (0.7) 4.00 (0.7) α heur f α GS f α GS f α GS f Matlab - Fluid: P -bubble/p - Structure: P

Numerical results II: simplified carotid Average number of iterations and (in brackets) relative CPU time α f = α s = 0 0 α P s α S s Ref. 3.08 (.00) 9.9 (0.76) 7.00 (0.54) 6.75 (0.5) H s = 0.05 3.5 (.00) 4.08 (0.6) 3.50 (0.58).67 (0.54) α heur f α GS f α GS f α GS f Matlab - Fluid: P -bubble/p - Structure: P

The Bidomain model for cardiac electrophysiology Complex problem: coexistence of intra- and extra-cellular media Ohm s Law charge conservation div (D i u i )=χ(c m t u + I ion (u, w)) in Ω (0, T ) div (D e u e )= χ(c m t u + I ion (u, w)) in Ω (0, T ) u = u i u e in Ω (0, T ) t w = R(u, w) in Ω (0, T ) u i intracellular potential u e extracellular potential w recovery variable n T D i,e u i,e = 0 in Ω (0, T ) u(x, 0) =u 0 (x), w(x, 0) =w 0 (x), in Ω Unique Up to a constant = Closing condition Ω u e dx = 0 MAIN ISSUE: Computational complexity

Simplifying Bidomain: Monodomain Model Bidomain in terms of u and u e χc m u t λdi + λ u [D i u +(D i + D e ) u e ]= I app. λdi D e + λ u e + I ion (u, w, c) =I app Monodomain λd i = D e D M (x) = λ + λ D i (x) χc m u t (D M (x) u) +I ion (u, w) =I app in Ω (0, T ) t w = R(u, w) in Ω (0, T ) n T D i u = 0 in Ω (0, T ) u(x, 0) =u 0 (x), w(x, 0) =w 0 (x), in Ω Pros Easier to solve Cons Less reliable... Model adaptivity: solve Bidomain only where needed = heterogeneous coupling

Bidomain-Monodomain coupling: Optimized Schwarz Coupling Bidomain/Bidomain coupling n T B D i u i,b + αu i,b = n T B D i u i,m + αu i,m n T B D e u e,b + βu e,b = n T B D e u e,m + βu e,m Non-symmetric formulation n T B λd i + λ ( u B + u e,b ) n T B = n T B D e + λ u e,b λd i + λ ( u M + u e,m ) n T B + λα + λ u B + λα β + λ u e,b D e + λ u e,m + λα + λ u M + λα β + λ u e,m n T B D i ( u B + u e,b )+n T B D e u e,b + αu B +(α + β)u e,b = n T B D i ( u M + u e,m )+n T B D e u e,m + αu M +(α + β)u e,m

Bidomain-Monodomain coupling: Optimized Schwarz Coupling Bidomain/Bidomain coupling n T B D i u i,b + αu i,b = n T B D i u i,m + αu i,m n T B D e u e,b + βu e,b = n T B D e u e,m + βu e,m Dimensional mismatch Non-symmetric formulation n T B λd i + λ ( u B + u e,b ) n T B = n T B D e + λ u e,b λd i + λ ( u M + u e,m ) n T B + λα + λ u B + λα β + λ u e,b D e + λ u e,m + λα + λ u M + λα β + λ u e,m n T B D i ( u B + u e,b )+n T B D e u e,b + αu B +(α + β)u e,b = n T B D i ( u M + u e,m )+n T B D e u e,m + αu M +(α + β)u e,m L.GG, M. Perego, A. Veneziani, Optimized Schwarz coupling of Bidomain and Monodomain models in electrocardiology, MAN, DOI: 0.05/man/00057, Published online: 0 Aug 00

Bidomain-Monodomain coupling: Optimized Schwarz Coupling Bidomain/Bidomain coupling n T B D i u i,b + αu i,b = n T B D i u i,m + αu i,m n T B D e u e,b + βu e,b = n T B D e u e,m + βu e,m Dimensional mismatch β = λα Non-symmetric formulation n T B λd i + λ ( u B + u e,b ) n T B = n T B D e + λ u e,b λd i + λ ( u M + u e,m ) n T B + λα + λ u B + λα β + λ u e,b D e + λ u e,m + λα + λ u M + λα β + λ u e,m n T B D i ( u B + u e,b )+n T B D e u e,b + αu B +(α + β)u e,b = n T B D i ( u M + u e,m )+n T B D e u e,m + αu M +(α + β)u e,m L.GG, M. Perego, A. Veneziani, Optimized Schwarz coupling of Bidomain and Monodomain models in electrocardiology, MAN, DOI: 0.05/man/00057, Published online: 0 Aug 00

Bidomain-Monodomain coupling: Optimized Schwarz Coupling Bidomain/Bidomain coupling n T B D i u i,b + αu i,b = n T B D i u i,m + αu i,m n T B D e u e,b + βu e,b = n T B D e u e,m + βu e,m Dimensional mismatch β = λα Non-symmetric formulation n T B λd i + λ ( u B + u e,b ) n T B = n T B D e + λ u e,b + λα + λ u B λd i + λ ( u M + u e,m ) n T B D e + λ u e,m + λα + λ u M n T B D i ( u B + u e,b )+n T B D e u e,b + αu B +( + λ)αu e,b = n T B D i ( u M + u e,m )+n T B D e u e,m + αu M +( + λ)αu e,m L.GG, M. Perego, A. Veneziani, Optimized Schwarz coupling of Bidomain and Monodomain models in electrocardiology, MAN, DOI: 0.05/man/00057, Published online: 0 Aug 00

Bidomain-Monodomain coupling: Optimized Schwarz Coupling Bidomain/Bidomain coupling n T B D i u i,b + αu i,b = n T B D i u i,m + αu i,m n T B D e u e,b + βu e,b = n T B D e u e,m + βu e,m Dimensional mismatch β = λα 3 λd i = D e Non-symmetric formulation n T B λd i + λ ( u B + u e,b ) n T B = n T B D e + λ u e,b + λα + λ u B λd i + λ ( u M + u e,m ) n T B D e + λ u e,m + λα + λ u M n T B D i ( u B + u e,b )+n T B D e u e,b + αu B +( + λ)αu e,b = n T B D i ( u M + u e,m )+n T B D e u e,m + αu M +( + λ)αu e,m L.GG, M. Perego, A. Veneziani, Optimized Schwarz coupling of Bidomain and Monodomain models in electrocardiology, MAN, DOI: 0.05/man/00057, Published online: 0 Aug 00

Bidomain-Monodomain coupling: Optimized Schwarz Coupling Bidomain/Bidomain coupling n T B D i u i,b + αu i,b = n T B D i u i,m + αu i,m n T B D e u e,b + βu e,b = n T B D e u e,m + βu e,m Dimensional mismatch β = λα 3 λd i = D e Non-symmetric formulation n T B λd i + λ ( u B + u e,b ) n T B D e + λ u e,b + λα + λ u B = n T B λd i + λ u M + λα + λ u M n T B D i ( u B + u e,b )+n T B D e u e,b + αu B +( + λ)αu e,b = n T B D i ( u M + u e,m )+n T B D e u e,m + αu M +( + λ)αu e,m L.GG, M. Perego, A. Veneziani, Optimized Schwarz coupling of Bidomain and Monodomain models in electrocardiology, MAN, DOI: 0.05/man/00057, Published online: 0 Aug 00

Bidomain-Monodomain coupling: Optimized Schwarz Coupling Bidomain/Bidomain coupling n T B D i u i,b + αu i,b = n T B D i u i,m + αu i,m n T B D e u e,b + βu e,b = n T B D e u e,m + βu e,m Dimensional mismatch β = λα 3 λd i = D e 4 Approx of u e,m : Non-symmetric formulation n T B λd i + λ ( u B + u e,b ) n T B D e + λ u e,b + λα + λ u B = n T B λd i + λ u M + λα + λ u M n T B D i ( u B + u e,b )+n T B D e u e,b + αu B +( + λ)αu e,b = n T B D i ( u M + u e,m )+n T B D e u e,m + αu M +( + λ)αu e,m L.GG, M. Perego, A. Veneziani, Optimized Schwarz coupling of Bidomain and Monodomain models in electrocardiology, MAN, DOI: 0.05/man/00057, Published online: 0 Aug 00

Bidomain-Monodomain coupling: Optimized Schwarz Coupling Bidomain/Bidomain coupling n T B D i u i,b + αu i,b = n T B D i u i,m + αu i,m n T B D e u e,b + βu e,b = n T B D e u e,m + βu e,m Dimensional mismatch β = λα 3 λd i = D e 4 Approx of u e,m : I app = 0, n T D e u e,m = 0, n T D i ( u M + u e,m )=0: u e,m = u M + K + λ Non-symmetric formulation n T B λd i + λ ( u B + u e,b ) n T B = n T B λd i + λ u M D e + λ u e,b + λα + λ u B + λα + λ u M n T B D i ( u B + u e,b )+n T B D e u e,b + αu B +( + λ)αu e,b = n T B D i ( u M + u e,m )+n T B D e u e,m + αu M +( + λ)αu e,m L.GG, M. Perego, A. Veneziani, Optimized Schwarz coupling of Bidomain and Monodomain models in electrocardiology, MAN, DOI: 0.05/man/00057, Published online: 0 Aug 00

Bidomain-Monodomain coupling: Optimized Schwarz Coupling Bidomain/Bidomain coupling n T B D i u i,b + αu i,b = n T B D i u i,m + αu i,m n T B D e u e,b + βu e,b = n T B D e u e,m + βu e,m Dimensional mismatch β = λα 3 λd i = D e 4 Approx of u e,m : I app = 0, n T D e u e,m = 0, n T D i ( u M + u e,m )=0: u e,m = u M + K + λ K = u rest + λ u e,m = 0 u M = u rest Non-symmetric formulation n T B λd i + λ ( u B + u e,b ) n T B = n T B λd i + λ u M D e + λ u e,b + λα + λ u B + λα + λ u M n T B D i ( u B + u e,b )+n T B D e u e,b + αu B +( + λ)αu e,b = n T B D i ( u M + u e,m )+n T B D e u e,m + αu M +( + λ)αu e,m L.GG, M. Perego, A. Veneziani, Optimized Schwarz coupling of Bidomain and Monodomain models in electrocardiology, MAN, DOI: 0.05/man/00057, Published online: 0 Aug 00

Bidomain-Monodomain coupling: Optimized Schwarz Coupling Bidomain/Bidomain coupling n T B D i u i,b + αu i,b = n T B D i u i,m + αu i,m n T B D e u e,b + βu e,b = n T B D e u e,m + βu e,m Dimensional mismatch β = λα 3 λd i = D e 4 Approx of u e,m : I app = 0, n T D e u e,m = 0, n T D i ( u M + u e,m )=0: u e,m = (u M u rest ) + λ Non-symmetric formulation n T B λd i + λ ( u B + u e,b ) n T B = n T B λd i + λ u M D e + λ u e,b + λα + λ u B + λα + λ u M n T B D i ( u B + u e,b )+n T B D e u e,b + αu B +( + λ)αu e,b = αu rest L.GG, M. Perego, A. Veneziani, Optimized Schwarz coupling of Bidomain and Monodomain models in electrocardiology, MAN, DOI: 0.05/man/00057, Published online: 0 Aug 00

Bidomain-Monodomain coupling: Optimized Schwarz Coupling Generalized Robin Coupling conditions on Γ n T B λd i + λ ( u B + u e ) n T B D e + λ u e + λα + λ u B = n T B λd i + λ u M + λα + λ u M n T B D i ( u B + u e )+n T B D e u e + αu B +( + λ)αu e = αu rest Convergence For α > 0, the OSA converges for every initial guess. Proof. Fourier analysis: the interface error reduces as e p+ Γ with ρ(α) <. ρ(α) e p Γ! 0.7 0.6 0.5 0.4 0.3 0. 0. 0 0 500 000 500 000 500 3000 k h =.e 3! 0.03 0.05 0.0 0.05 0.0 0.005 0 0 50 00 50 00 50 300 k h =.e L.GG, M. Perego, A. Veneziani, Optimized Schwarz coupling of Bidomain and Monodomain models in electrocardiology, MAN, DOI: 0.05/man/00057, Published online: 0 Aug 00

Bidomain-Monodomain coupling: Convergence Convergence (L.GG, M. Perego, A. Veneziani, MAN, 0) For α > 0, the Optimized Schwarz Algorithm converges for every initial guess, and for every Fourier mode k its reduction factor is independent of the iteration and is given by α σi l µm (k) α [ Ψ(k, α)] σ ρ(k, α) = α + σi l µm (k) i l η Ψ(k, α) (k ) η + (k ) α [ Ψ(k, α)] + σi l η Ψ(k, α) (k, ) η + (k ) where µ M (k) = χc m t σ l i + σ l e + σt i σt e σ t i + σ t e σ l i + σ l e σ l i σl e k, Ψ(k, α) = (σt i + σe t )k (σi l + σe l ) η (k) (σi t + σe)k t (σi l + σe) l η + (k) α + σl i η + (k) α + σi l η (k), and η ± (k) = χcm σi l t ± σ l i + σi t k + σe l χcm t χcm t + σi t k σe l + σ t ek χcm t + σ t ek + 4 σ l i σl e χcm. t

Bidomain-Monodomain coupling: Parameter Optimization Exact solution: ρ(α exact ) 0 α exact (k) = σi l µm (k) =σi l χc m t σ l i + σ l e + σt i σt e σ t i + σ t e σ l i + σ l e σ l i σl e k unpractical α exact (k) : α exact (k) = σ l i η + (k) Ψ(k, α exact (k)) η (k) Ψ(k, α exact (k)) η + (k) Ideal solution (still difficult to compute) min α R + max k [0,k max ] ρ(k, α) Viable choice: low frequency approximation α opt = α exact (0) α = σ l i χc m t σ l i +σl e σ l i σl e