NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS

By CHARLOTTE SMITH

CONTENTS FOR EXAMPLE: MODELLING PROCESSES IN POROUS MEDIA WITH DI?ERENTIAL EQUATIONS THE BASIC PARTIAL DI?ERENTIAL EQUATION MODELS REACTIONS AND TRANSPORT IN POROUS MEDIA FLUID FLOW IN POROUS MEDIA REACTIVE SOLUTE TRANSPORT IN POROUS MEDIA BOUNDARY AND INITIAL VALUE PROBLEMS FOR THE BEGINNING: THE FINITE DI?ERENCE METHOD FOR THE POISSON EQUATION THE DIRICHLET PROBLEM FOR THE POISSON EQUATION THE FINITE DI?ERENCE METHOD GENERALIZATIONS AND LIMITATIONS OF THE FINITE DI?ERENCE METHOD MAXIMUM PRINCIPLES AND STABILITY THE FINITE ELEMENT METHOD FOR THE POISSON EQUATION VARIATIONAL FORMULATION FOR THE MODEL PROBLEM THE FINITE ELEMENT METHOD WITH LINEAR ELEMENTS STABILITY AND CONVERGENCE OF THE FINITE ELEMENT METHOD THE IMPLEMENTATION OF THE FINITE ELEMENT METHOD: PART SOLVING SPARSE SYSTEMS OF LINEAR EQUATIONS BY DIRECT METHODS THE FINITE ELEMENT METHOD FOR LINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS OF SECOND ORDER VARIATIONAL EQUATIONS AND SOBOLEV SPACES ELLIPTIC BOUNDARY VALUE PROBLEMS OF SECOND ORDER ELEMENT TYPES AND A?NE EQUIVALENT TRIANGULATIONS CONVERGENCE RATE ESTIMATES CONVERGENCE RATE RESULTS IN CASE OF QUADRATURE AND INTERPOLATION THE CONDITION NUMBER OF FINITE ELEMENT MATRICES GENERAL DOMAINS AND ISOPARAMETRIC ELEMENTS THE MAXIMUM PRINCIPLE FOR FINITE ELEMENT METHODS GRID GENERATION AND A POSTERIORI ERROR ESTIMATION GRID GENERATION A POSTERIORI ERROR ESTIMATES AND GRID ADAPTATION ITERATIVE METHODS FOR SYSTEMS OF LINEAR EQUATIONS LINEAR STATIONARY ITERATIVE METHODS GRADIENT AND CONJUGATE GRADIENT METHODS PRECONDITIONED CONJUGATE GRADIENT METHOD KRYLOV SUBSPACE METHODS FOR NONSYMMETRIC SYSTEMS OF EQUATIONS THE MULTIGRID METHOD NESTED ITERATIONS THE FINITE VOLUME METHOD THE BASIC IDEA OF THE FINITE VOLUME METHOD THE FINITE VOLUME METHOD FOR LINEAR ELLIPTIC DI?EREN- TIAL EQUATIONS OF SECOND ORDER ON TRIANGULAR GRIDS DISCRETIZATION METHODS FOR PARABOLIC INITIAL BOUNDARY VALUE PROBLEMS PROBLEM SETTING AND SOLUTION CONCEPT SEMIDISCRETIZATION BY THE VERTICAL METHOD OF LINES FULLY DISCRETE SCHEMES STABILITY THE MAXIMUM PRINCIPLE FOR THE ONE-STEP-THETA METHOD ORDER OF CONVERGENCE ESTIMATES ITERATIVE METHODS FOR NONLINEAR EQUATIONS FIXED-POINT ITERATIONS NEWTON’S METHOD AND ITS VARIANTS SEMILINEAR BOUNDARY VALUE PROBLEMS FOR ELLIPTIC AND PARABOLIC EQUATIONS DISCRETIZATION METHODS FOR CONVECTION-DOMINATED PROBLEMS STANDARD METHODS AND CONVECTION-DOMINATED PROBLEMS THE STREAMLINE-DI?USION METHOD FINITE VOLUME METHODS THE LAGRANGE–GALERKIN METHOD REFERENCES: TEXTBOOKS AND MONOGRAPHS REFERENCES: JOURNAL PAPERS