ELLIPTIC PROBLEM SOLVERS
CONTENTS PART I ANALYTICAL RESULTS FROM LINEAR TO NONLINEAR EQUATIONS, FUNDAMENTAL RESULTS INTRODUCTION LINEAR VERSUS NONLINEAR MODELS EXAMPLES FOR NONLINEAR PARTIAL DI?ERENTIAL EQUATIONS FUNDAMENTAL RESULTS LINEAR OPERATORS AND FUNCTIONALS IN BANACH SPACES INEQUALITIES AND LP(?) SPACES H¨OLDER AND SOBOLEV SPACES AND MORE DERIVATIVES IN BANACH SPACES ELEMENTS OF ANALYSIS FOR LINEAR AND NONLINEAR PARTIAL ELLIPTIC DI?ERENTIAL EQUATIONS AND SYSTEMS LINEAR ELLIPTIC DI?ERENTIAL OPERATORS OF SECOND ORDER, BILINEAR FORMS AND SOLUTION CONCEPTS BILINEAR FORMS AND INDUCED LINEAR OPERATORS LINEAR ELLIPTIC DI?ERENTIAL OPERATORS, FREDHOLM ALTERNATIVE AND REGULAR SOLUTIONS LINEAR OPERATORS OF ORDER 2M WITH C?COE?CIENTS LINEAR OPERATORS OF ORDER 2 UNDER CK CONDITIONS WEAK ELLIPTIC EQUATION OF ORDER 2M IN HILBERT SPACES NONLINEAR ELLIPTIC EQUATIONS DE?NITIONS FOR NONLINEAR ELLIPTIC OPERATORS SPECIAL SEMILINEAR AND QUASILINEAR OPERATORS QUASILINEAR ELLIPTIC EQUATIONS OF ORDER GENERAL NONLINEAR AND NEMYCKII OPERATORS DIVERGENT QUASILINEAR ELLIPTIC EQUATIONS OF ORDER 2M FULLY NONLINEAR ELLIPTIC EQUATIONS OF ORDERS 2, M AND 2M LINEAR AND NONLINEAR ELLIPTIC SYSTEMS GENERAL SYSTEMS OF ELLIPTIC DI?ERENTIAL EQUATIONS LINEAR ELLIPTIC SYSTEMS OF ORDER QUASILINEAR ELLIPTIC SYSTEMS OF ORDER 2 AND VARIATIONAL METHODS DIVERGENT QUASILINEAR ELLIPTIC SYSTEMS OF ORDER 2M NEMYCKII OPERATORS AND QUASILINEAR DIVERGENT SYSTEMS OF ORDER 2M FULLY NONLINEAR ELLIPTIC SYSTEMS OF ORDERS 2 AND 2M LINEARIZATION OF NONLINEAR OPERATORS SPECIAL SEMILINEAR AND QUASILINEAR EQUATIONS DIVERGENT QUASILINEAR AND FULLY NONLINEAR EQUATIONS QUASILINEAR ELLIPTIC SYSTEMS OF ORDERS 2 AND 2M LINEARIZING GENERAL DIVERGENT QUASILINEAR AND FULLY NONLINEAR SYSTEMS THE NAVIER–STOKES EQUATION THE STOKES OPERATOR AND SADDLE POINT PROBLEMS THE NAVIER–STOKES OPERATOR AND ITS LINEARIZATION PART II NUMERICAL METHODS A GENERAL DISCRETIZATION THEORY PETROV–GALERKIN AND GENERAL DISCRETIZATION METHODS VARIATIONAL AND CLASSICAL CONSISTENCY STABILITY AND CONSISTENCY YIELD CONVERGENCE TECHNIQUES FOR PROVING STABILITY STABILITY IMPLIES INVERTIBILITY SOLVING NONLINEAR SYSTEMS: CONTINUATION AND NEWTON’S METHOD BASED UPON THE MESH INDEPENDENCE PRINCIPLE (MIP) CONTINUATION METHODS MIP FOR NONLINEAR SYSTEMS CONFORMING ?NITE ELEMENT METHODS (FEMS) APPROXIMATION THEORY FOR ?NITE ELEMENTS SUBDIVISIONS AND ?NITE ELEMENTS POLYNOMIAL ?NITE ELEMENTS, TRIANGULAR AND RECTANGULAR K INTERPOLATION IN ?NITE ELEMENT SPACES, AN EXAMPLE INTERPOLATION ERRORS AND INVERSE ESTIMATES INVERSE ESTIMATES ON NONQUASIUNIFORM TRIANGULATIONS SMOOTH FES ON POLYHEDRAL DOMAINS, WITH O DAVYDOV CURVED BOUNDARIES FEMS FOR LINEAR PROBLEMS FINITE ELEMENT METHODS: A SIMPLE EXAMPLE, ESSENTIAL TOOLS FINITE ELEMENT METHODS FOR GENERAL LINEAR EQUATIONS AND SYSTEMS OF ORDERS 2 AND 2M GENERAL CONVERGENCE THEORY FOR CONFORMING FEMS FINITE ELEMENT METHODS FOR DIVERGENT QUASILINEAR ELLIPTIC EQUATIONS AND SYSTEMS GENERAL CONVERGENCE THEORY FOR MONOTONE AND QUASILINEAR OPERATORS MIXED FEMS FOR NAVIER–STOKES AND SADDLE POINT EQUATIONS NAVIER–STOKES AND SADDLE POINT EQUATIONS MIXED FEMS FOR STOKES AND SADDLE POINT EQUATIONS MIXED FEMS FOR THE NAVIER–STOKES OPERATOR VARIATIONAL METHODS FOR EIGENVALUE PROBLEMS THEORY FOR EIGENVALUE PROBLEMS DI?ERENT VARIATIONAL METHODS FOR EIGENVALUE PROBLEMS NONCONFORMING ?NITE ELEMENT METHODS FINITE ELEMENT METHODS FOR FULLY NONLINEAR ELLIPTIC PROBLEMS MAIN IDEAS AND RESULTS FOR THE NEW FEM: AN EXTENDED SUMMARY FULLY NONLINEAR AND GENERAL QUASILINEAR ELLIPTIC EQUATIONS EXISTENCE AND CONVERGENCE FOR SEMICONFORMING FEMS DE?NITION OF NONCONFORMING FEMS CONSISTENCY FOR NONCONFORMING FEMS STABILITY FOR THE LINEARIZED OPERATOR AND CONVERGENCE DISCRETIZATION OF EQUATIONS AND SYSTEMS OF ORDER 2M 5 2 10 NUMERICAL SOLUTION OF THE FE EQUATIONS WITH NEWTON’S METHOD FE AND OTHER METHODS FOR NONLINEAR BOUNDARY CONDITIONS QUADRATURE APPROXIMATE FEMS QUADRATURE AND CUBATURE FORMULAS QUADRATURE FOR SECOND ORDER LINEAR PROBLEMS QUADRATURE FOR SECOND ORDER FULLY NONLINEAR EQUATIONS QUADRATURE FEMS FOR EQUATIONS AND SYSTEMS OF ORDER 2M TWO USEFUL PROPOSITIONS CONSISTENCY, STABILITY AND CONVERGENCE FOR FEMS WITH VARIATIONAL CRIMES VARIATIONAL CRIMES FOR OUR STANDARD EXAMPLE FEMS WITH CRIMES FOR LINEAR AND QUASILINEAR PROBLEMS DISCRETE COERCIVITY AND CONSISTENCY HIGH ORDER QUADRATURE ON EDGES VIOLATED BOUNDARY CONDITIONS VIOLATED CONTINUITY STABILITY FOR NONCONFORMING FEMS CONVERGENCE, QUADRATURE AND SOLUTION OF FEMS WITH CRIMES 5 5 10 ISOPARAMETRIC FEMS ADAPTIVE ?NITE ELEMENT METHODS, BY W D¨OR?ER THE MODEL PROBLEM SINGULAR SOLUTIONS A PRIORI ERROR BOUNDS NECESSITY OF NONUNIFORM MESH RE?NEMENT OPTIMAL MESHES – A HEURISTIC ARGUMENT OPTIMAL MESHES FOR 2D CORNER SINGULARITIES THE ?NITE ELEMENT METHOD–NOTATION AND REQUIREMENTS THE RESIDUAL ERROR ESTIMATOR FOR THE POISSON PROBLEM UPPER A POSTERIORI BOUND LOWER A POSTERIORI BOUND THE A POSTERIORI ERROR ESTIMATE THE ADAPTIVE ?NITE ELEMENT METHOD CONVERGENCE OF THE ADAPTIVE ?NITE ELEMENT METHOD OPTIMALITY OTHER TYPES OF ESTIMATORS HP ?NITE ELEMENT METHOD ESTIMATION OF QUANTITIES OF INTEREST QUANTITIES OF INTEREST ERROR ESTIMATES FOR POINT ERRORS OPTIMAL MESHES–A HEURISTIC ARGUMENT THE GENERAL APPROACH DISCONTINUOUS GALERKIN METHODS (DCGMS), WITH V DOLEJ?S´I DISCRETIZATION OF THE PROBLEM TRIANGULATIONS BROKEN SOBOLEV SPACES EXTENDED VARIATIONAL FORMULATION OF THE PROBLEM DISCRETIZATION GENERAL LINEAR ELLIPTIC PROBLEMS SEMILINEAR AND QUASILINEAR ELLIPTIC PROBLEMS SEMILINEAR ELLIPTIC PROBLEMS VARIATIONAL FORMULATION AND DISCRETIZATION OF THE PROBLEM QUASILINEAR ELLIPTIC SYSTEMS DISCRETIZATION OF THE QUASILINEAR SYSTEMS DCGMS ARE GENERAL DISCRETIZATION METHODS GEOMETRY OF THE MESH, ERROR AND INVERSE ESTIMATES GEOMETRY OF THE MESH INVERSE AND INTERPOLATION ERROR ESTIMATES PENALTY NORMS AND CONSISTENCY OF THE J? H COERCIVE LINEARIZED PRINCIPAL PARTS COERCIVITY OF THE ORIGINAL LINEARIZED PRINCIPAL PARTS COERCIVITY AND BOUNDEDNESS IN VH FOR THE LAPLACIAN COERCIVITY AND BOUNDEDNESS IN VH FOR THE GENERAL LINEAR AND THE SEMILINEAR CASE VH-COERCIVITY AND BOUNDEDNESS FOR QUASILINEAR PROBLEMS 7 10 CONSISTENCY RESULTS FOR THE CH, BH, ?H 7 10 1 CONSISTENCY OF THE CH AND BH 7 10 2 CONSISTENCY OF THE ?H 7 11 CONSISTENCY PROPERTIES OF THE AH 7 11 1 CONSISTENCY OF THE AH FOR THE LAPLACIAN 7 11 2 CONSISTENCY OF THE AH FOR GENERAL LINEAR PROBLEMS 7 11 3 CONSISTENCY OF THE SEMILINEAR AH 7 11 4 CONSISTENCY OF THE QUASILINEAR AH FOR SYSTEMS 7 11 5 CONSISTENCY OF THE QUASILINEAR AH FOR THE EQUATIONS OF HOUSTON, ROBSON, S¨ULI, AND FOR SYSTEMS 7 12 CONVERGENCE FOR DCGMS 7 13 SOLVING NONLINEAR EQUATIONS IN DCGMS 7 13 1 INTRODUCTION 7 13 2 DISCRETIZED LINEARIZED QUASILINEAR SYSTEM AND DI?ERENTIABLE CONSISTENCY 7 14 HP-VARIANTS OF DCGM 7 14 1 HP-?NITE ELEMENT SPACES 7 14 2 HP-DCGMS 7 14 3 HP-INVERSE AND APPROXIMATION ERROR ESTIMATES 7 14 4 CONSISTENCY AND CONVERGENCE OF HP-DCGMS 7 15 NUMERICAL EXPERIENCES 7 15 1 SCALAR QUASILINEAR EQUATION 7 15 2 SYSTEM OF THE STEADY COMPRESSIBLE NAVIER–STOKES EQUATIONS FINITE DI?ERENCE METHODS DI?ERENCE METHODS FOR SIMPLE EXAMPLES, NOTATION DISCRETE SOBOLEV SPACES NOTATION AND DE?NITIONS GENERAL ELLIPTIC PROBLEMS WITH DIRICHLET CONDITIONS, AND THEIR DI?ERENCE METHODS GENERAL ELLIPTIC PROBLEMS SECOND ORDER LINEAR ELLIPTIC DI?ERENCE EQUATIONS SYMMETRIC DI?ERENCE METHODS LINEAR EQUATIONS OF ORDER 2M QUASILINEAR ELLIPTIC EQUATIONS OF ORDERS 2, AND 2M SYSTEMS OF LINEAR AND QUASILINEAR ELLIPTIC EQUATIONS FULLY NONLINEAR ELLIPTIC EQUATIONS AND SYSTEMS CONVERGENCE FOR DI?ERENCE METHODS DISCRETIZATION CONCEPTS IN DISCRETE SOBOLEV SPACES THE OPERATORS P H, Q ?H CONSISTENCY FOR DI?ERENCE EQUATIONS VH B -COERCIVITY FOR LINEAR(IZED) ELLIPTIC DI?ERENCE EQUATIONS STABILITY AND CONVERGENCE FOR GENERAL ELLIPTIC DI?ERENCE EQUATIONS NATURAL BOUNDARY VALUE PROBLEMS OF ORDER ANALYSIS FOR NATURAL BOUNDARY VALUE PROBLEMS DI?ERENCE METHODS FOR NATURAL BOUNDARY VALUE PROBLEMS OTHER DI?ERENCE METHODS ON CURVED BOUNDARIES THE SHORTLEY–WELLER–COLLATZ METHOD FOR LINEAR EQUATIONS ASYMPTOTIC EXPANSIONS, EXTRAPOLATION, AND DEFECT CORRECTIONS A DI?ERENCE METHOD BASED ON POLYNOMIAL INTERPOLATION FOR LINEAR, AND SEMILINEAR EQUATIONS ASYMPTOTIC EXPANSIONS FOR OTHER METHODS NUMERICAL EXPERIMENTS FOR THE VON K´ARM´AN EQUATIONS, WITH C S CHIEN VARIATIONAL METHODS FOR WAVELETS, WITH S DAHLKE THE SCOPE OF PROBLEMS WAVELET ANALYSIS THE DISCRETE WAVELET TRANSFORM BIORTHOGONAL BASES WAVELETS AND FUNCTION SPACES WAVELETS ON DOMAINS EVALUATION OF NONLINEAR FUNCTIONALS STABLE DISCRETIZATIONS AND PRECONDITIONING APPLICATIONS TO ELLIPTIC EQUATIONS SADDLE POINT AND (NAVIER–)STOKES EQUATIONS SADDLE POINT EQUATIONS NAVIER–STOKES EQUATIONS ADAPTIVE WAVELET METHODS, BY T RAASCH NONLINEAR APPROXIMATION WITH WAVELET SYSTEMS WAVELET MATRIX COMPRESSION ADAPTIVE WAVELET–GALERKIN METHODS ADAPTIVE DESCENT ITERATIONS NONLINEAR STATIONARY PROBLEMS BIBLIOGRAPHY