Outline Coupled FETI/BETI Solvers for Nonlinear Potential Problems in Unbounded Domains Clemens Pechstein 1 Ulrich Langer 1 , 2 1 Special Research Programm SFB F013 on Numerical and Symbolic Scientific Computing 2 Institute of Computational Mathematics Johannes Kepler University Linz DD 17, Strobl/Wolfgangsee, 2006 Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains
Outline Outline FETI/BETI Introduction 1 Motivation and Overview Coupled FETI/BETI Formulation Preconditioners Generalizations 2 Unbounded Domains Nonlinear Problems Conclusion 3 Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains
Introduction Motivation and Overview Generalizations Coupled FETI/BETI Formulation Conclusion Preconditioners Outline FETI/BETI Introduction 1 Motivation and Overview Coupled FETI/BETI Formulation Preconditioners Generalizations 2 Unbounded Domains Nonlinear Problems Conclusion 3 Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains
Introduction Motivation and Overview Generalizations Coupled FETI/BETI Formulation Conclusion Preconditioners Motivation – Electromagnetic Field Computations Nonlinear Magnetostatics in 2D: � � − ∇ · ν ( |∇ u | ) ∇ u = f + boundary or radiation conditions + transmission conditions Characteristics: Nonlinear in ferromagnetic materials Linear behavior in surrounding air / air gaps Typically large jumps over material interfaces Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains
Introduction Motivation and Overview Generalizations Coupled FETI/BETI Formulation Conclusion Preconditioners Motivation – Electromagnetic Field Computations Nonlinear Magnetostatics in 2D: � � − ∇ · ν ( |∇ u | ) ∇ u = f + boundary or radiation conditions + transmission conditions Characteristics: Nonlinear in ferromagnetic materials Linear behavior in surrounding air / air gaps Typically large jumps over material interfaces Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains
Introduction Motivation and Overview Generalizations Coupled FETI/BETI Formulation Conclusion Preconditioners F inite E lement T earing and I nterconnecting – Overview Farhat and Roux, 1991 Non-overlapping Domain Decomposition b.v.p. for Poisson Problem, Structural Mechanics Domain Decomposition Conformal mesh Separate d.o.f. Continuity → Lagrange multipliers Elimination → dual problem PCG sub-space iteration Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains
Introduction Motivation and Overview Generalizations Coupled FETI/BETI Formulation Conclusion Preconditioners F inite E lement T earing and I nterconnecting – Overview Farhat and Roux, 1991 Non-overlapping Domain Decomposition b.v.p. for Poisson Problem, Structural Mechanics Domain Decomposition Conformal mesh Separate d.o.f. Continuity → Lagrange multipliers Elimination → dual problem PCG sub-space iteration Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains
Introduction Motivation and Overview Generalizations Coupled FETI/BETI Formulation Conclusion Preconditioners F inite E lement T earing and I nterconnecting – Overview Farhat and Roux, 1991 Non-overlapping Domain Decomposition b.v.p. for Poisson Problem, Structural Mechanics Domain Decomposition Conformal mesh Separate d.o.f. Continuity → Lagrange multipliers Elimination → dual problem PCG sub-space iteration Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains
Introduction Motivation and Overview Generalizations Coupled FETI/BETI Formulation Conclusion Preconditioners F inite E lement T earing and I nterconnecting – Overview Farhat and Roux, 1991 Non-overlapping Domain Decomposition b.v.p. for Poisson Problem, Structural Mechanics Domain Decomposition Conformal mesh Separate d.o.f. Continuity → Lagrange multipliers Elimination → dual problem PCG sub-space iteration Tearing Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains
Introduction Motivation and Overview Generalizations Coupled FETI/BETI Formulation Conclusion Preconditioners F inite E lement T earing and I nterconnecting – Overview Farhat and Roux, 1991 Non-overlapping Domain Decomposition b.v.p. for Poisson Problem, Structural Mechanics Domain Decomposition Conformal mesh Separate d.o.f. Continuity → Lagrange multipliers Elimination → dual problem PCG sub-space iteration Interconnecting Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains
Introduction Motivation and Overview Generalizations Coupled FETI/BETI Formulation Conclusion Preconditioners F inite E lement T earing and I nterconnecting – Overview Farhat and Roux, 1991 Non-overlapping Domain Decomposition b.v.p. for Poisson Problem, Structural Mechanics Domain Decomposition Conformal mesh Separate d.o.f. Continuity → Lagrange multipliers Elimination → dual problem PCG sub-space iteration Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains
Introduction Motivation and Overview Generalizations Coupled FETI/BETI Formulation Conclusion Preconditioners F inite E lement T earing and I nterconnecting – Overview Farhat and Roux, 1991 Non-overlapping Domain Decomposition b.v.p. for Poisson Problem, Structural Mechanics Domain Decomposition Conformal mesh Separate d.o.f. Continuity → Lagrange multipliers Elimination → dual problem PCG sub-space iteration Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains
Introduction Motivation and Overview Generalizations Coupled FETI/BETI Formulation Conclusion Preconditioners F inite E lement T earing and I nterconnecting – Overview FETI – Features PCG iteration and preconditioning via local Dirichlet and Neumann solvers Allows massive parallelization Mandel/Tezaur, 1996 Condition number O (( 1 + log ( H / h )) 2 ) Klawonn/Widlund, Robust w.r.t. coefficient jumps 2001 Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains
Introduction Motivation and Overview Generalizations Coupled FETI/BETI Formulation Conclusion Preconditioners B oundary E lement T earing and I nterconnecting – Overview FETI-technique can be carried over to the BEM → BETI Langer/Steinbach, 2003 Coupling BETI and FETI: Benetit from advantages of both techniques air gaps / surrounding air → BEM nonlinear materials → FEM Langer/Steinbach, 2004 Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains
Introduction Motivation and Overview Generalizations Coupled FETI/BETI Formulation Conclusion Preconditioners B oundary E lement T earing and I nterconnecting – Overview FETI-technique can be carried over to the BEM → BETI Langer/Steinbach, 2003 Coupling BETI and FETI: Benetit from advantages of both techniques air gaps / surrounding air → BEM nonlinear materials → FEM Langer/Steinbach, 2004 Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains
Introduction Motivation and Overview Generalizations Coupled FETI/BETI Formulation Conclusion Preconditioners Outline FETI/BETI Introduction 1 Motivation and Overview Coupled FETI/BETI Formulation Preconditioners Generalizations 2 Unbounded Domains Nonlinear Problems Conclusion 3 Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains
Introduction Motivation and Overview Generalizations Coupled FETI/BETI Formulation Conclusion Preconditioners A Continuous Formulation Quasi-regular non-overlapping Domain Decomposition Ω ⊂ R d bounded Γ = ∂ Ω Ω = � i ∈I Ω i Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains
Introduction Motivation and Overview Generalizations Coupled FETI/BETI Formulation Conclusion Preconditioners A Continuous Formulation Quasi-regular non-overlapping Domain Decomposition Ω ⊂ R d bounded Γ = ∂ Ω Ω = � i ∈I Ω i Γ i = ∂ Ω i n i : outward unit normal vector Γ jk = Γ j ∩ Γ k Γ S = � i ∈I Γ i H i = diam Ω i ≃ H Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains
Introduction Motivation and Overview Generalizations Coupled FETI/BETI Formulation Conclusion Preconditioners A Continuous Formulation Quasi-regular non-overlapping Domain Decomposition Ω ⊂ R d bounded Γ = ∂ Ω Ω = � i ∈I Ω i Γ i = ∂ Ω i n i : outward unit normal vector Γ jk = Γ j ∩ Γ k Γ S = � i ∈I Γ i H i = diam Ω i ≃ H Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains
Introduction Motivation and Overview Generalizations Coupled FETI/BETI Formulation Conclusion Preconditioners A Continuous Formulation Quasi-regular non-overlapping Domain Decomposition Ω ⊂ R d bounded Γ = ∂ Ω Ω = � i ∈I Ω i Γ i = ∂ Ω i n i : outward unit normal vector Γ jk = Γ j ∩ Γ k Γ S = � i ∈I Γ i H i = diam Ω i ≃ H Clemens Pechstein, Ulrich Langer FETI/BETI – Nonlinear – Unbounded Domains
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