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
CONTENTS 1 The Mathematical Foundation of Domain Decomposition Methods 1 1.1 Multi-domain formulation and the Steklov-Poincare interface equation 2 1.2 Variational formulation of the multi-domain problem 5 1.3 Iterative substructuring methods based on transmission conditions at the interface 10 1.4 Generalisations 18 1.4.1 The Steklov-Poincare equation for the Neumann boundary value problem 22 1.4.2 Iterations on many subdomains 24 1.5 The Schwarz method for overlapping subdomains 26 1.5.1 The multiplicative and additive forms of the Schwarz method 26 1.5.2 Variational Interpretation of the Schwarz method 28 1.5.3 The Schwarz method as a projection method 29 1.5.4 The Schwarz method as a Richardson method 31 1.5.5 A characterisation of the projection Operators 32 1.5.6 The Schwarz method for many subdomains 33 1.6 The fictitious domain method 34 1.7 The three-field method 38 2 Discretised Equations and Domain Decomposition Methods 41 2.1 Finite element approximation of elliptic equations 41 2.1.1 The multi-domain formulation for finite elements 43 2.1.2 Algebraic formulation of the discrete problem 45 2.2 Finite element approximation of the Steklov-Poincare Operator 46 2.2.1 Eigenvalue analysis for the finite element Steklov- Poincare Operator 48 2.3 Algebraic formulation of the discrete Steklov-Poincare Operator: the Schur complement matrix 49 2.3.1 Preconditioners of the stiffness matrix derived from preconditioners of the Schur complement matrix 51 2.4 The case of many subdomains 55
Xll CONTENTS 2.5 Non-conforming domain decomposition methods 59 2.5.1 The mortar method 60 2.5.2 The three-field method at the finite dimensional level 66 3 Iterative Domain Decomposition Methods at the Discrete Level 71 3.1 Iterative substructuring methods at the finite element level 71 3.2 The link between the Schur complement System and iterative substructuring methods 73 3.3 Schur complement preconditioners 77 3.3.1 Decomposition with two subdomains 77 3.3.2 Decomposition with many subdomains 79 3.4 The Schwarz method for finite elements 86 3.5 Acceleration of the Schwarz method 91 3.5.1 Inexact solvers 96 3.6 Two-level methods 96 3.6.1 Abstract setting of two-level methods 97 3.6.2 Multiplicative and additive two-level preconditioners 98 3.6.3 The case of the Schwarz method 99 3.6.4 Convergence of two-level methods 99 3.7 Direct Galerkin approximation of the Steklov-Poincare equation 100 4 Convergence Analysis for Iterative Domain Decomposition Algorithms 103 4.1 Extension theorems and spectrally equivalent Operators 104 4.1.1 Extension theorems in ij 1 (ft i ) 104 4.1.2 Extension theorems in ij(div; fij) 111 4.1.3 Extension theorems in H(rot; fl^) 114 4.2 Splitting of Operators and preconditioned iterative methods 117 4.2.1 The finite dimensional case 122 4.2.2 The case of Symmetrie matrices 125 4.2.3 The case of complex matrices 128 4.3 Convergence of the Dirichlet-Neumann iterative method 133 4.3.1 An alternative way to prove convergence 133 4.4 Convergence of the Neumann-Neumann iterative method 135 4.5 Convergence of the Robin iterative method 135 4.6 Convergence of the alternating Schwarz method 137 5 Other Boundary Value Problems 141 5.1 Non-symmetric elliptic Operators 141
CONTENTS xm 5.1.1 Weak multi-domain formulation and the Steklov- Poincare interface equation 142 5.1.2 Substructuring iterative methods 146 5.1.3 The finite dimensional approximation 147 5.2 The problem of linear elasticity 147 5.2.1 Weak multi-domain formulation and the Steklov- Poincare interface equation 148 5.2.2 Substructuring iterative methods 151 5.2.3 The finite dimensional approximation 151 5.3 The Stokes problem 153 5.3.1 Weak multi-domain formulation and the Steklov- Poincare interface equation 156 5.3.2 Substructuring iterative methods 170 5.3.3 Finite dimensional approximation: the case of discontinuous pressure 173 5.3.4 Finite dimensional approximation: the case of continuous pressure 185 5.3.5 Methods based on the Uzawa pressure Operator 189 5.4 The Stokes problem for compressible flows 190 5.4.1 Weak multi-domain formulation and the Steklov- Poincare interface equation 191 5.4.2 Substructuring iterative methods 194 5.4.3 The finite dimensional approximation 195 5.5 The Stokes problem for inviscid compressible flows 197 5.5.1 Weak multi-domain formulation and the Steklov- Poincare interface equation 198 5.5.2 Substructuring iterative methods 201 5.5.3 The finite dimensional approximation 202 5.6 First-order equations 203 5.6.1 Weak multi-domain formulation and the Steklov- Poincare interface equation 204 5.6.2 Substructuring iterative methods 208 5.7 The time-harmonic Maxwell equations 210 5.7.1 Weak multi-domain formulation 211 5.7.2 The finite dimensional Steklov-Poincare interface equation 213 5.7.3 Substructuring iterative methods 214 6 Advection Diffusion Equations 219 6.1 The advection-diffusion problem and its multi-domain formulations 220 6.2 Iterative substructuring methods for one-dimensional Problems 224 6.3 Adaptive iterative substructuring methods: ADN, ARN and AR^N 227
XIV CONTENTS 6.3.1 The damped form of the iterative algorithms: d- ADN, d-arn and d-ar^n 232 6.4 Coercive iterative substructuring methods: 7-DR and 7- RR 234 6.4.1 The 7-DR iterative method 234 6.4.2 Convergence of the 7-DR iterative method 237 6.4.3 The 7-DR iterative method for Systems of advection-diffusion equations 239 6.4.4 The 7-RR iterative method 240 6.4.5 Convergence of the 7-RR iterative method 242 6.5 The finite element realisation of the iterative algorithms 244 7 Time-Dependent Problems 251 7.1 Parabolic problems 252 7.1.1 Multi-domain formulation and space discretisation 253 7.1.2 Implicit time discretisation and subdomain iterations 256 7.2 Hyperbolic problems 261 7.2.1 Multi-domain formulation 263 7.2.2 Implicit time discretisation and subdomain iterations 266 7.3 Non-linear time-dependent problems 272 7.3.1 Navier-Stokes equations for incompressible flows 273 7.3.2 Navier-Stokes equations for compressible flows 275 7.3.3 Euler equations for compressible flows 277 8 Heterogeneous Domain Decomposition Methods 285 8.1 Heterogeneous modeis for advection-diffusion equations 287 8.1.1 The Steklov-Poincare reformulation 290 8.1.2 The coupling for non-linear convection-diffusion equations 295 8.2 Heterogeneous modeis for incompressible flows 296 8.2.1 The coupling for the linearised Stokes problem 297 8.2.2 The coupling for the Navier-Stokes equations in exterior domains 300 8.3 Heterogeneous modeis for compressible flows 305 8.3.1 The coupling between the Navier-Stokes and Euler equations 306 8.3.2 The coupling between the Euler equations and the füll potential equation 309 8.4 The coupling for the compressible Stokes equations 317 8.4.1 Variational formulation and finite element approximation 320 8.4.2 An alternative formulation: variational setting and finite element approximation 322
CONTENTS 8.5 The coupling for the time-harmonic Maxwell equations 328 9 Appendix 333 9.1 Function Spaces 333 9.2 Some properties of the Sobolev Spaces 339 References 343 Index 357