Domain Decomposition Methods. Partial Differential Equations

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1 Domain Decomposition Methods for Partial Differential Equations ALFIO QUARTERONI Professor ofnumericalanalysis, Politecnico di Milano, Italy, and Ecole Polytechnique Federale de Lausanne, Switzerland ALBERTO VALLI Professor offunctionalanalysis, University oftrento, Italy CLARENDON PRESS OXFORD 1999

2 CONTENTS 1 The Mathematical Foundation of Domain Decomposition Methods Multi-domain formulation and the Steklov-Poincare interface equation Variational formulation of the multi-domain problem Iterative substructuring methods based on transmission conditions at the interface Generalisations The Steklov-Poincare equation for the Neumann boundary value problem Iterations on many subdomains The Schwarz method for overlapping subdomains The multiplicative and additive forms of the Schwarz method Variational Interpretation of the Schwarz method The Schwarz method as a projection method The Schwarz method as a Richardson method A characterisation of the projection Operators The Schwarz method for many subdomains The fictitious domain method The three-field method 38 2 Discretised Equations and Domain Decomposition Methods Finite element approximation of elliptic equations The multi-domain formulation for finite elements Algebraic formulation of the discrete problem Finite element approximation of the Steklov-Poincare Operator Eigenvalue analysis for the finite element Steklov- Poincare Operator Algebraic formulation of the discrete Steklov-Poincare Operator: the Schur complement matrix Preconditioners of the stiffness matrix derived from preconditioners of the Schur complement matrix The case of many subdomains 55

3 Xll CONTENTS 2.5 Non-conforming domain decomposition methods The mortar method The three-field method at the finite dimensional level 66 3 Iterative Domain Decomposition Methods at the Discrete Level Iterative substructuring methods at the finite element level The link between the Schur complement System and iterative substructuring methods Schur complement preconditioners Decomposition with two subdomains Decomposition with many subdomains The Schwarz method for finite elements Acceleration of the Schwarz method Inexact solvers Two-level methods Abstract setting of two-level methods Multiplicative and additive two-level preconditioners The case of the Schwarz method Convergence of two-level methods Direct Galerkin approximation of the Steklov-Poincare equation Convergence Analysis for Iterative Domain Decomposition Algorithms Extension theorems and spectrally equivalent Operators Extension theorems in ij 1 (ft i ) Extension theorems in ij(div; fij) Extension theorems in H(rot; fl^) Splitting of Operators and preconditioned iterative methods The finite dimensional case The case of Symmetrie matrices The case of complex matrices Convergence of the Dirichlet-Neumann iterative method An alternative way to prove convergence Convergence of the Neumann-Neumann iterative method Convergence of the Robin iterative method Convergence of the alternating Schwarz method Other Boundary Value Problems Non-symmetric elliptic Operators 141

4 CONTENTS xm Weak multi-domain formulation and the Steklov- Poincare interface equation Substructuring iterative methods The finite dimensional approximation The problem of linear elasticity Weak multi-domain formulation and the Steklov- Poincare interface equation Substructuring iterative methods The finite dimensional approximation The Stokes problem Weak multi-domain formulation and the Steklov- Poincare interface equation Substructuring iterative methods Finite dimensional approximation: the case of discontinuous pressure Finite dimensional approximation: the case of continuous pressure Methods based on the Uzawa pressure Operator The Stokes problem for compressible flows Weak multi-domain formulation and the Steklov- Poincare interface equation Substructuring iterative methods The finite dimensional approximation The Stokes problem for inviscid compressible flows Weak multi-domain formulation and the Steklov- Poincare interface equation Substructuring iterative methods The finite dimensional approximation First-order equations Weak multi-domain formulation and the Steklov- Poincare interface equation Substructuring iterative methods The time-harmonic Maxwell equations Weak multi-domain formulation The finite dimensional Steklov-Poincare interface equation Substructuring iterative methods Advection Diffusion Equations The advection-diffusion problem and its multi-domain formulations Iterative substructuring methods for one-dimensional Problems Adaptive iterative substructuring methods: ADN, ARN and AR^N 227

5 XIV CONTENTS The damped form of the iterative algorithms: d- ADN, d-arn and d-ar^n Coercive iterative substructuring methods: 7-DR and 7- RR The 7-DR iterative method Convergence of the 7-DR iterative method The 7-DR iterative method for Systems of advection-diffusion equations The 7-RR iterative method Convergence of the 7-RR iterative method The finite element realisation of the iterative algorithms Time-Dependent Problems Parabolic problems Multi-domain formulation and space discretisation Implicit time discretisation and subdomain iterations Hyperbolic problems Multi-domain formulation Implicit time discretisation and subdomain iterations Non-linear time-dependent problems Navier-Stokes equations for incompressible flows Navier-Stokes equations for compressible flows Euler equations for compressible flows Heterogeneous Domain Decomposition Methods Heterogeneous modeis for advection-diffusion equations The Steklov-Poincare reformulation The coupling for non-linear convection-diffusion equations Heterogeneous modeis for incompressible flows The coupling for the linearised Stokes problem The coupling for the Navier-Stokes equations in exterior domains Heterogeneous modeis for compressible flows The coupling between the Navier-Stokes and Euler equations The coupling between the Euler equations and the füll potential equation The coupling for the compressible Stokes equations Variational formulation and finite element approximation An alternative formulation: variational setting and finite element approximation 322

6 CONTENTS 8.5 The coupling for the time-harmonic Maxwell equations Appendix Function Spaces Some properties of the Sobolev Spaces 339 References 343 Index 357

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