# Domain Decomposition Methods in Scientific and Engineering Computing

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*David E. Keyes; Jinchao Xu*

This book contains proceedings from the Seventh International Conference on Domain Decomposition Methods, held at Pennsylvania State University in October 1993. The term “domain decomposition” has for nearly a decade been associated with the partly iterative, partly direct algorithms explored in the proceedings of this conference. Noteworthy trends in the current volume include progress in dealing with so-called “bad parameters” in elliptic partial differential equation problems, as well as developments in partial differential equations outside of the elliptically-dominated framework. Also described here are convergence and complexity results for novel discretizations, which bring with them new challenges in the derivation of appropriate operators for coarsened spaces. Implementations and architectural considerations are discussed, as well as partitioning tools and environments. In addition, the book describes a wide array of applications, from semiconductor device simulation to structural mechanics to aerodynamics. Presenting many of the latest results in the field, this book offers readers an up-to-date guide to the many facets of the theory and practice of domain decomposition.

# Table of Contents

## Domain Decomposition Methods in Scientific and Engineering Computing

- Contents v6 free
- Preface xi12 free
- List of participant's xix20 free
- Part I. Theory 130 free
- Interpolat'ion spaces and optimal multilevel preconditioners 332
- Two-level additive Schwarz preconditioners for nonconforming finite elements 938
- Two proofs of convergence for the combination technique for the efficient solution of sparse grid problems 1544
- Domain decomposition rnethods for monotone nonlinear elliptic problems 2150
- Cascadic conjugate gradient methods for elliptic partial differential equations: Algorithm and numerical results 2958
- Multilevel methods for elliptic problems with discontinuous coefficients in three dimensions 4372
- Multilevel methods for elliptic problems on domains not resolved by the coarse grid 4978
- Three-dimensional domain decomposition methods with nonmatching grids and unstructured coarse solvers 6190
- Domain decomposition for elliptic problems with large condition numbers 75104
- Stable subspace splittings for Sobolev spaces and domain decomposition algorithms 87116
- A wire basket based method for spectral elements in three dimensions 99128
- An analysis of spectral graph partitioning via quadratic assignment problems 105134
- Error estimators based on stable splittings 111140
- Multilevel methods for PI nonconforming finite elements and discontinuous coefficients in three dimensions 119148
- On generalized Schwarz coupling applied to advection-dominated problems 125154
- Exotic coarse spaces for Schwarz methods for lower order and spectral finite elements 131160

- Part II. Algorithms 137166
- Preconditioning via asymptotically-defined domain decomposition 139168
- A spectral Stokes solver in domain decomposition methods 151180
- Preconditioned iterative methods in a subspace for linear algebraic equations with large jumps in the coefficients 157186
- The hierarchical basis multigrid method and incomplete LU decomposition 163192
- Domain decomposition and multigrid algorithms for elliptic problems on unstructured meshes 175204
- Two-grid methods for mixed finite element approximations of nonlinear parabolic equations 191220
- Domain decomposition algorithms for PDE problems with large scale variations 205234
- A one shot domain decomposition/fictitious domain method for the Navier-Stokes equations 211240
- Domain-oriented multilevel methods 223252
- Multigrid and domain decomposition methods for electrostatics problems 231260
- A parallel subspace decomposition method for hyperbolic equations 239268
- Numerical treatments for the Helmholtz problem by domain decomposition techniques 245274
- Schwarz methods for obstacle problems with convection-diffusion operators 251280
- On domain decomposition and shooting methods for two-point boundary value problems 257286
- Robust methods for highly nonsymmetric problems 265294
- Domain decomposition via the Sinc-Galerkin method for second order differential equations 271300
- A bisection method to find all solutions of a system of nonlinear equations 277306
- Preconditioning cell-centered finite difference equations on grids with local refinement 283312
- Outflow boundary conditions and domain decomposition method 289318
- Domain decomposition for adaptive hp finite element methods 295324
- Domain decomposed preconditioners with Krylov subspace methods as subdomain solver 303332
- Elliptic precondltioaes using fast summation techniques 311340
- The Modified vertex space domain decomposition method for Neumann boundary value problems 325354
- A multi-color splitting rnethod and convergerice analysis for local grid refinement 337366
- Boundary elements in domain decomposition methods 343372
- An overdetermined Schwarz alternating method 349378
- Domain decomposition for linear and nonlinear elliptic problems via function or space decomposition 355384
- ELLAM-based domain decomposition and local refinement techniques for advection-diffusion equations with interfaces 361390

- Part III. Parallelism 367396
- Parallel domain decomposition applied to coupled transport equations 369398
- MENUS-PGG: A mapping environment for unstructured and structured numerical parallel grid generation 381410
- A comparison of three iterative algorithms based on domain decomposition methods 387416
- IBLU preconditioners for massively parallel computers 395424
- Tailoring domain decomposition methods for efficient parallel coarse grid solution and for systems with many right hand sides 401430
- Analysis and implementation of DD methods for parallel FE computations 407436
- Finite-element/Newton method for solution of nonlinear problems in transport processes using domain decomposition and nested dissection on MIMD parallel computers 413442
- Modeling with collaborating PDE solvers: Theory and practice 427456
- Parallelization of a multigrid solver via a domain decomposition method 439468
- A compiler for parallel finite element methods with domain-decomposed unstructured meshes 445474

- Part IV. Applications 451480
- A X-formulation of the viscous-inviscid domain decomposition for the Euler/Navier-Stokes equations 453482
- Domain decomposition and computation of two dimensional detonation waves 459488
- Parallel implicit methods for aerodynamics 465494
- Parallel domain-decomposed preconditioners in finite element shallow water flow modeling 471500
- A domain decomposition method for Bellman equations 477506
- Domain decomposition for the shallow water equations 485514
- Domain decomposition methods for device modelling 491520
- An efficient computational method for the flow past an airfoil 497526
- Newton-Krylov-Schwarz techniques applied to the two-dimensional incompressible Navier-Stokes and energy equations 503532
- Direct numerical simulation of jet flow via a multi-block technique 509538
- Balancing domain decomposition for plates 515544
- Nonlinear block iterative solution of semiconductor device equations by a domain decomposition method 525554
- A direct Chebyshev multidomain method for flow computation with application to rotating systems 533562
- Multi-domain Fourier algorithms for parallel solution of the Navier-Stokes equations 539568