The Finite Element Method

By KENNETH ROBINSON

CONTENTS 0 FOR EXAMPLE MODELLING PROCESSES IN POROUS MEDIA WITH DI ERENTIAL EQUATIONS 1 0 1 THE BASIC PARTIAL DI ERENTIAL EQUATION MODELS 1 0 2 REACTIONS AND TRANSPORT IN POROUS MEDIA 5 0 3 FLUID FLOW IN POROUS MEDIA 7 0 4 REACTIVE SOLUTE TRANSPORT IN POROUS MEDIA 11 0 5 BOUNDARY AND INITIAL VALUE PROBLEMS 14 1 FOR THE BEGINNING THE FINITE DI ERENCE METHOD FOR THE POISSON EQUATION 19 1 1 THE DIRICHLET PROBLEM FOR THE POISSON EQUATION 19 1 2 THE FINITE DI ERENCE METHOD 21 1 3 GENERALIZATIONS AND LIMITATIONS OF THE FINITE DI ERENCE METHOD 29 1 4 MAXIMUM PRINCIPLES AND STABILITY 36 2 THE FINITE ELEMENT METHOD FOR THE POISSON EQUATION 46 2 1 VARIATIONAL FORMULATION FOR THE MODEL PROBLEM 46 2 2 THE FINITE ELEMENT METHOD WITH LINEAR ELEMENTS 55 2 3 STABILITY AND CONVERGENCE OF THE FINITE ELEMENT METHOD 68 2 4 THE IMPLEMENTATION OF THE FINITE ELEMENT METHOD PART 1 74 2 5 SOLVING SPARSE SYSTEMS OF LINEAR EQUATIONS BY DIRECT METHODS 82 3 THE FINITE ELEMENT METHOD FOR LINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS OF SECOND ORDER 92 3 1 VARIATIONAL EQUATIONS AND SOBOLEV SPACES 92 3 2 ELLIPTIC BOUNDARY VALUE PROBLEMS OF SECOND ORDER 100 3 3 ELEMENT TYPES AND A NE EQUIVALENT TRIANGULATIONS 114 3 4 CONVERGENCE RATE ESTIMATES 131 3 5 THE IMPLEMENTATION OF THE FINITE ELEMENT METHOD PART 2 148 3 6 CONVERGENCE RATE RESULTS IN CASE OF QUADRATURE AND INTERPOLATION 155 3 7 THE CONDITION NUMBER OF FINITE ELEMENT MATRICES 163 3 8 GENERAL DOMAINS AND ISOPARAMETRIC ELEMENTS 167 3 9 THE MAXIMUM PRINCIPLE FOR FINITE ELEMENT METHODS 171 4 GRID GENERATION AND A POSTERIORI ERROR ESTIMATION 176 4 1 GRID GENERATION 176 4 2 A POSTERIORI ERROR ESTIMATES AND GRID ADAPTATION 185 5 ITERATIVE METHODS FOR SYSTEMS OF LINEAR EQUATIONS 198 5 1 LINEAR STATIONARY ITERATIVE METHODS 200 5 2 GRADIENT AND CONJUGATE GRADIENT METHODS 217 5 3 PRECONDITIONED CONJUGATE GRADIENT METHOD 227 5 4 KRYLOV SUBSPACE METHODS FOR NONSYMMETRIC SYSTEMS OF EQUATIONS 233 5 5 THE MULTIGRID METHOD 238 5 6 NESTED ITERATIONS 251 6 THE FINITE VOLUME METHOD 255 6 1 THE BASIC IDEA OF THE FINITE VOLUME METHOD 256 6 2 THE FINITE VOLUME METHOD FOR LINEAR ELLIPTIC DI EREN TIAL EQUATIONS OF SECOND ORDER ON TRIANGULAR GRIDS 262 7 DISCRETIZATION METHODS FOR PARABOLIC INITIAL BOUNDARY VALUE PROBLEMS 283 7 1 PROBLEM SETTING AND SOLUTION CONCEPT 283 7 2 SEMIDISCRETIZATION BY THE VERTICAL METHOD OF LINES 293