Spectral analysis and preconditioning in Finite Element approximations of elliptic PDEs Cristina Tablino-Possio joint work with Alessandro Russo Dipartimento di Matematica e Applicazioni, Universit` a di Milano-Bicocca, Milano, Italy Structured Linear Algebra Problems: Analysis, Algorithms, and Applications Cortona, September 15-19, 2008 C. Tablino-Possio (Universit` a di Milano-Bicocca) Spectral analysis and preconditioning in FE approximations of elliptic PDEs Cortona 2008 1 / 28
Outline Problem 1 Motivations 2 PHSS Method 3 Preconditioning Strategy 4 Spectral Analysis 5 Complexity Issues 6 Numerical Tests 7 Some Perspectives & Conclusions 8 C. Tablino-Possio (Universit` a di Milano-Bicocca) Spectral analysis and preconditioning in FE approximations of elliptic PDEs Cortona 2008 2 / 28
Problem The Problem - Variational Form Convection-Diffusion Equations � � � − a ( x ) ∇ u + � β ( x ) u = f , x ∈ Ω , div u | ∂ Ω = 0 . Variational form find u ∈ H 1 0 (Ω) such that � � � � a ∇ u · ∇ ϕ − � for all ϕ ∈ H 1 β · ∇ ϕ u = f ϕ 0 (Ω) . Ω Ω Regularity Assumptions a ∈ C 2 (Ω) , with a ( x ) ≥ a 0 > 0 , � with div ( � β ∈ C 1 (Ω) , β ) ≥ 0 pointwise in Ω , f ∈ L 2 (Ω) . C. Tablino-Possio (Universit` a di Milano-Bicocca) Spectral analysis and preconditioning in FE approximations of elliptic PDEs Cortona 2008 3 / 28
Problem The Problem - FE Approximation - I Let T h = { K } finite element partition of Ω, polygonal domain, into triangles, h K = diam ( K ), h = max K h K . We consider the space of linear finite elements V h = { ϕ h : Ω → R s.t. ϕ h is continuous, ϕ h | K is linear, and ϕ h | ∂ Ω = 0 }⊂ H 1 0 (Ω) with basis ϕ i ∈ V h s.t. ϕ i (node j ) = δ i , j , i , j = 1 , . . . , n ( h ) , n ( h ) = dim ( V h ) = number of the internal nodes of T h . C. Tablino-Possio (Universit` a di Milano-Bicocca) Spectral analysis and preconditioning in FE approximations of elliptic PDEs Cortona 2008 4 / 28
Problem The Problem - FE Approximation - II The variational equation becomes A n ( a , � β ) u = b T h = { K } with A n ( a , � � n ( a , � Θ n ( a ) + Ψ n ( � A K β ) ∈ R n × n , β ) = β ) = n = n ( h ) , K ∈T h � � (Θ n ( a )) i , j = a ∇ ϕ j · ∇ ϕ i diffusive term , K K ∈T h � (Ψ n ( � � ( � β )) i , j = − β · ∇ ϕ i ) ϕ j convective term , K K ∈T h and with suitable quadrature formulas in the case of non constant a and � β . C. Tablino-Possio (Universit` a di Milano-Bicocca) Spectral analysis and preconditioning in FE approximations of elliptic PDEs Cortona 2008 5 / 28
Motivations Motivations Motivations Recent attention to Hermitian/Skew-Hermitian splitting (HSS) iterations proposed in Bai et al. (2003) for non-Hermitian linear systems with positive definite real part: Bai, Benzi, Bertaccini, Gander, Golub, Ng, Serra-Capizza- no, Simoncini, TP, . . . Preconditioned HHS splitting iterations proposed in Bertaccini et al. (2005) for non-Hermitian linear systems with positive definite real part. Previously considered preconditioning strategy for FD/FE approximations of diffusion Eqns and FD approximations of Convection-Diffusion Eqns: Beckermann, Bertaccini, Golub, Serra-Capizzano, TP, . . . Aim To study the effectiveness of the proposed Preconditioned HSS method applied to the FE approximations of Convection-Diffusion Eqns. both from the theoretical and numerical point of view. C. Tablino-Possio (Universit` a di Milano-Bicocca) Spectral analysis and preconditioning in FE approximations of elliptic PDEs Cortona 2008 6 / 28
Motivations Motivations Motivations Recent attention to Hermitian/Skew-Hermitian splitting (HSS) iterations proposed in Bai et al. (2003) for non-Hermitian linear systems with positive definite real part: Bai, Benzi, Bertaccini, Gander, Golub, Ng, Serra-Capizza- no, Simoncini, TP, . . . Preconditioned HHS splitting iterations proposed in Bertaccini et al. (2005) for non-Hermitian linear systems with positive definite real part. Previously considered preconditioning strategy for FD/FE approximations of diffusion Eqns and FD approximations of Convection-Diffusion Eqns: Beckermann, Bertaccini, Golub, Serra-Capizzano, TP, . . . Aim To study the effectiveness of the proposed Preconditioned HSS method applied to the FE approximations of Convection-Diffusion Eqns. both from the theoretical and numerical point of view. C. Tablino-Possio (Universit` a di Milano-Bicocca) Spectral analysis and preconditioning in FE approximations of elliptic PDEs Cortona 2008 6 / 28
PHSS Method PHSS Method - Definition Let us consider A n ∈ C n × n with a positive definite real part , x , b ∈ C n A n x = b , The HSS method [1] refers to the Hermitian/Skew-Hermitian splitting i 2 = − 1 A n = Re ( A n ) + i Im ( A n ) , with Re ( A n ) = ( A n + A H n ) / 2 and Im ( A n ) = ( A n − A H n ) / (2 i ) Hermitian matrices. Here, we consider the Preconditioned HSS (PHSS) method [2] x k + P − 1 x k + 1 � � α I + P − 1 � � α I − P − 1 � n Re ( A n ) = n i Im ( A n ) n b 2 x k + 1 � α I + P − 1 � x k +1 � α I − P − 1 � 2 + P − 1 n i Im ( A n ) = n Re ( A n ) n b with P n Hermitian positive definite matrix and α positive parameter. [1] Bai, Golub, Ng, SIMAX, 2003. [2] Bertaccini, Golub, Serra-Capizzano, TP, Numer. Math., 2005. C. Tablino-Possio (Universit` a di Milano-Bicocca) Spectral analysis and preconditioning in FE approximations of elliptic PDEs Cortona 2008 7 / 28
PHSS Method PHSS Method - Convergence Properties Theorem (Bertaccini et al. , 2005) Let A n ∈ C n × n be a matrix with positive definite real part, let α be a positive parameter and let P n ∈ C n × n be a Hermitian positive definite matrix. Then, the PHSS method is unconditionally convergent, since � � α − λ i � � ̺ ( M ( α )) ≤ σ ( α ) = max � < 1 for any α > 0 , � � α + λ i λ i ∈ λ ( P − 1 Re ( A n )) � n with iteration matrix � − 1 � � − 1 � α I + i P − 1 α I − P − 1 α I + P − 1 α I − i P − 1 � � � � M ( α ) = Im ( A n ) Re ( A n ) Re ( A n ) Im ( A n ) . n n n n Moreover, the optimal α value that minimizes the quantity σ ( α ) equals √ κ − 1 � α ∗ = λ min ( P − 1 n Re ( A n )) λ max ( P − 1 n Re ( A n )) and σ ( α ∗ ) = √ κ + 1 with κ = λ max ( P − 1 n Re ( A n )) /λ min ( P − 1 n Re ( A n )) spectral condition number. C. Tablino-Possio (Universit` a di Milano-Bicocca) Spectral analysis and preconditioning in FE approximations of elliptic PDEs Cortona 2008 8 / 28
PHSS Method PHSS/IPHSS Method - Inexact Iterations From a practical point of view, the PHSS method can also be interpreted as the original HSS method where the identity matrix is replaced by P n , i.e., ( α P n − i Im ( A n )) x k + b ( α P n + Re ( A n )) x k + 1 � = 2 ( α P n − Re ( A n )) x k + 1 ( α P n + i Im ( A n )) x k +1 2 + b . = In practice, the two half-steps of the outer iteration can be computed by applying a PCG and a Preconditioned GMRES method, with preconditioner P n . The accuracy for the stopping criterion of these additional inner iterative procedures is chosen by taking into account the accuracy obtained by the current step of the outer iteration. We denote by IPHSS method the described inexact PHSS iterations. C. Tablino-Possio (Universit` a di Milano-Bicocca) Spectral analysis and preconditioning in FE approximations of elliptic PDEs Cortona 2008 9 / 28
Preconditioning Strategy The Preconditioning Strategy - Definition - I In the case of the considered FE approximation of Convection-Diffusion Eqns, the Hermitian/skew-Hermitian splitting is given by Re ( A n ( a , � � n ( a , � β )) = Θ n ( a ) + Re (Ψ n ( � Re ( A K β )) = β )) spd , K ∈T h i Im ( A n ( a , � � n ( a , � β )) = i Im (Ψ n ( � Im ( A K β )) = i β )) , K ∈T h and can be performed on any single elementary matrix related to T h . Notice that Re (Ψ n ( � β )) = 0 if div ( � β ) = 0. Lemma Let { E n ( � E n ( � β ) := Re (Ψ n ( � β ) } , β )) . Under the regularity assumptions, then it holds � E n ( � β ) � 2 ≤ � E n ( � β ) � ∞ ≤ Ch 2 , with C absolute constant only depending on � β ( x ) and Ω . C. Tablino-Possio (Universit` a di Milano-Bicocca) Spectral analysis and preconditioning in FE approximations of elliptic PDEs Cortona 2008 10 / 28
Preconditioning Strategy The Preconditioning Strategy - Definition - II The considered preconditioning matrix sequence, proposed in [1], is defined as 1 1 { P n ( a ) } , P n ( a ) = D n ( a ) A n (1 , 0) D 2 n ( a ) 2 where D n ( a ) = diag ( A n ( a , 0)) diag − 1 ( A n (1 , 0)), i.e., the suitable scaled main diagonal of A n ( a , 0) and A n ( a , 0) equals Θ n ( a ). Notice that the preconditioner is tuned only with respect to the diffusion matrix Θ n ( a ) owing to the PHSS convergence properties. [1] Serra-Capizzano, Numer. Math., 1999. C. Tablino-Possio (Universit` a di Milano-Bicocca) Spectral analysis and preconditioning in FE approximations of elliptic PDEs Cortona 2008 11 / 28
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