THE MATHEMATICAL THEORY OF FINITE ELEMENT METHODS

By LINDA WILLIAMS

CONTENTS 0 BASIC CONCEPTS 0 1 WEAK FORMULATION OF BOUNDARY VALUE PROBLEMS 0 2 RITZ-GALERKIN APPROXIMATION 0 3 ERROR ESTIMATES 0 4 PIECEWISE POLYNOMIAL SPACES – THE FINITE ELEMENT METHOD 0 5 RELATIONSHIP TO DI?ERENCE METHODS 0 6 COMPUTER IMPLEMENTATION OF FINITE ELEMENT METHODS 0 7 LOCAL ESTIMATES 0 8 ADAPTIVE APPROXIMATION 0 9 WEIGHTED NORM ESTIMATES 0 X EXERCISES 1 SOBOLEV SPACES 1 1 REVIEW OF LEBESGUE INTEGRATION THEORY 1 2 GENERALIZED (WEAK) DERIVATIVES 1 3 SOBOLEV NORMS AND ASSOCIATED SPACES 1 4 INCLUSION RELATIONS AND SOBOLEV’S INEQUALITY 1 5 REVIEW OF CHAPTER 1 6 TRACE THEOREMS 1 7 NEGATIVE NORMS AND DUALITY 1 X EXERCISES 2 VARIATIONAL FORMULATION OF ELLIPTIC BOUNDARY VALUE PROBLEMS 2 1 INNER-PRODUCT SPACES 2 2 HILBERT SPACES 2 3 PROJECTIONS ONTO SUBSPACES 2 4 RIESZ REPRESENTATION THEOREM 2 5 FORMULATION OF SYMMETRIC VARIATIONAL PROBLEMS 2 6 FORMULATION OF NONSYMMETRIC VARIATIONAL PROBLEMS 2 7 THE LAX-MILGRAM THEOREM 2 8 ESTIMATES FOR GENERAL FINITE ELEMENT APPROXIMATION 2 9 HIGHER-DIMENSIONAL EXAMPLES 2 X EXERCISES 3 THE CONSTRUCTION OF A FINITE ELEMENT SPACE 3 1 THE FINITE ELEMENT 3 2 TRIANGULAR FINITE ELEMENTS THE LAGRANGE ELEMENT THE HERMITE ELEMENT THE ARGYRIS ELEMENT 3 3 THE INTERPOLANT 3 4 EQUIVALENCE OF ELEMENTS 3 5 RECTANGULAR ELEMENTS TENSOR PRODUCT ELEMENTS THE SERENDIPITY ELEMENT 3 6 HIGHER–DIMENSIONAL ELEMENTS 3 7 EXOTIC ELEMENTS 3 X EXERCISES 4 POLYNOMIAL APPROXIMATION THEORY IN SOBOLEV SPACES 4 1 AVERAGED TAYLOR POLYNOMIALS 4 2 ERROR REPRESENTATION 4 3 BOUNDS FOR RIESZ POTENTIALS 4 4 BOUNDS FOR THE INTERPOLATION ERROR 4 5 INVERSE ESTIMATES 4 6 TENSOR-PRODUCT POLYNOMIAL APPROXIMATION 4 7 ISOPARAMETRIC POLYNOMIAL APPROXIMATION 4 8 INTERPOLATION OF NON-SMOOTH FUNCTIONS 4 9 A DISCRETE SOBOLEV INEQUALITY 4 X EXERCISES 5 N-DIMENSIONAL VARIATIONAL PROBLEMS 5 1 VARIATIONAL FORMULATION OF POISSON’S EQUATION 5 2 VARIATIONAL FORMULATION OF THE PURE NEUMANN PROBLEM 5 3 COERCIVITY OF THE VARIATIONAL PROBLEM 5 4 VARIATIONAL APPROXIMATION OF POISSON’S EQUATION 5 5 ELLIPTIC REGULARITY ESTIMATES 5 6 GENERAL SECOND-ORDER ELLIPTIC OPERATORS 5 7 VARIATIONAL APPROXIMATION OF GENERAL ELLIPTIC PROBLEMS 5 8 NEGATIVE-NORM ESTIMATES 5 9 THE PLATE-BENDING BIHARMONIC PROBLEM 5 X EXERCISES 6 FINITE ELEMENT MULTIGRID METHODS 6 1 A MODEL PROBLEM 6 2 MESH-DEPENDENT NORMS 6 3 THE MULTIGRID ALGORITHM 6 4 APPROXIMATION PROPERTY 6 5 W-CYCLE CONVERGENCE FOR THE KTH LEVEL ITERATION 6 6 V-CYCLE CONVERGENCE FOR THE KTH LEVEL ITERATION 6 7 FULL MULTIGRID CONVERGENCE ANALYSIS AND WORK ESTIMATES 6 X EXERCISES 7 ADDITIVE SCHWARZ PRECONDITIONERS 7 1 ABSTRACT ADDITIVE SCHWARZ FRAMEWORK 7 2 THE HIERARCHICAL BASIS PRECONDITIONER 7 3 THE BPX PRECONDITIONER 7 4 THE TWO-LEVEL ADDITIVE SCHWARZ PRECONDITIONER 7 5 NONOVERLAPPING DOMAIN DECOMPOSITION METHODS 7 6 THE BPS PRECONDITIONER 7 7 THE NEUMANN-NEUMANN PRECONDITIONER 7 8 THE BDDC PRECONDITIONER 7 X EXERCISES 8 MAX–NORM ESTIMATES 8 1 MAIN THEOREM 8 2 REDUCTION TO WEIGHTED ESTIMATES 8 4 PROOFS OF LEMMAS 8 3 7 AND 8 3 11 8 5 LP ESTIMATES (REGULAR COE?CIENTS) 8 6 LP ESTIMATES (IRREGULAR COE?CIENTS) 8 7 A NONLINEAR EXAMPLE 8 X EXERCISES 9 ADAPTIVE MESHES 9 1 A PRIORI ESTIMATES 9 2 ERROR ESTIMATORS 9 3 LOCAL ERROR ESTIMATES 9 4 ESTIMATORS FOR LINEAR FORMS AND OTHER NORMS 9 5 A CONVERGENT ADAPTIVE ALGORITHM 9 6 CONDITIONING OF FINITE ELEMENT EQUATIONS 9 7 BOUNDS ON THE CONDITION NUMBER 9 8 APPLICATIONS TO THE CONJUGATE-GRADIENT METHOD 9 X EXERCISES 10 VARIATIONAL CRIMES 10 1 DEPARTURE FROM THE FRAMEWORK 10 2 FINITE ELEMENTS WITH INTERPOLATED BOUNDARY CONDITIONS 10 3 NONCONFORMING FINITE ELEMENTS 10 4 ISOPARAMETRIC FINITE ELEMENTS 10 5 DISCONTINUOUS FINITE ELEMENTS 10 6 POINCAR´E-FRIEDRICHS INEQUALITITES FOR PIECEWISE W P FUNCTIONS 10 X EXERCISES 11 APPLICATIONS TO PLANAR ELASTICITY 11 1 THE BOUNDARY VALUE PROBLEMS 11 2 WEAK FORMULATION AND KORN’S INEQUALITY 11 3 FINITE ELEMENT APPROXIMATION AND LOCKING 11 4 A ROBUST METHOD FOR THE PURE DISPLACEMENT PROBLEM 11 X EXERCISES 12 MIXED METHODS 12 1 EXAMPLES OF MIXED VARIATIONAL FORMULATIONS 12 2 ABSTRACT MIXED FORMULATION 12 3 DISCRETE MIXED FORMULATION 12 4 CONVERGENCE RESULTS FOR VELOCITY APPROXIMATION 12 5 THE DISCRETE INF-SUP CONDITION 12 6 VERI?CATION OF THE INF-SUP CONDITION 12 X EXERCISES 13 ITERATIVE TECHNIQUES FOR MIXED METHODS 13 1 ITERATED PENALTY METHOD 13 2 STOPPING CRITERIA 13 3 AUGMENTED LAGRANGIAN METHOD 13 4 APPLICATION TO THE NAVIER-STOKES EQUATIONS 13 5 COMPUTATIONAL EXAMPLES 13 X EXERCISES 14 APPLICATIONS OF OPERATOR-INTERPOLATION THEORY 14 1 THE REAL METHOD OF INTERPOLATION 14 2 REAL INTERPOLATION OF SOBOLEV SPACES 14 3 FINITE ELEMENT CONVERGENCE ESTIMATES 14 4 THE SIMULTANEOUS APPROXIMATION THEOREM 14 5 PRECISE CHARACTERIZATIONS OF REGULARITY 14 X EXERCISES REFERENCES