A POSTERIORI ERROR ESTIMATION IN FINITE ELEMENT ANALYSIS

By O. GAARA

CONTENTS A SIMPLE MODEL PROBLEM 1 1 MOTIVATION AND OVERVIEW 1 2 THE MODEL PROBLEM AND ITS DISCRETISATION 1 3 NOTATIONS AND AUXILIARY RESULTS 1 4 RESIDUAL ESTIMATES 1 5 A VERTEX-ORIENTED RESIDUAL ERROR INDICATOR 1 6 EDGE RESIDUALS 1 7 AUXILIARY LOCAL PROBLEMS 1 8 A HIERARCHICAL APPROACH 1 9 GRADIENT RECOVERY 1 10 EQUILIBRATED RESIDUALS 1 11 DUAL WEIGHTED RESIDUALS 1 12 THE HYPER-CIRCLE METHOD 1 13 EF?CIENCY AND ASYMPTOTIC EXACTNESS 1 14 CONVERGENCE OF THE ADAPTIVE PROCESS I 1 15 SUMMARY AND OUTLOOK IMPLEMENTATION 2 1 MESH-RE?NEMENT 2 2 MESH-COARSENING 2 3 MESH-SMOOTHING 2 4 DATA STRUCTURES 2 5 NUMERICAL EXAMPLES AUXILIARY RESULTS 3 1 FUNCTION SPACES 3 2 FINITE ELEMENT MESHES AND SPACES 3 3 TRACE INEQUALITIES 3 4 POINCARÉ AND FRIEDRICHS’ INEQUALITIES 3 5 INTERPOLATION ERROR ESTIMATES 3 6 INVERSE ESTIMATES 3 7 DECOMPOSITION OF AF?NE FUNCTIONS IN LP(0, 1; Y?) 3 8 ESTIMATION OF RESIDUALS LINEAR ELLIPTIC EQUATIONS 4 1 ABSTRACT LINEAR PROBLEMS 4 2 THE MODEL PROBLEM REVISITED 4 3 REACTION–DIFFUSION EQUATIONS 4 4 CONVECTION–DIFFUSION EQUATIONS 4 5 ANISOTROPIC MESHES 4 6 NON-SMOOTH COEF?CIENTS 4 7 EIGENVALUE PROBLEMS 4 8 MIXED FORMULATION OF THE POISSON EQUATION 4 9 THE EQUATIONS OF LINEAR ELASTICITY 4 10 THE STOKES EQUATIONS 4 11 THE BI-HARMONIC EQUATION 4 12 NON–CONFORMING DISCRETISATIONS 4 13 CONVERGENCE OF THE ADAPTIVE PROCESS II NONLINEAR ELLIPTIC EQUATIONS 5 1 ABSTRACT NONLINEAR PROBLEMS 5 2 QUASILINEAR EQUATIONS OF SECOND ORDER 5 3 EIGENVALUE PROBLEMS REVISITED 5 4 THE STATIONARY NAVIER–STOKES EQUATIONS PARABOLIC EQUATIONS 6 1 THE HEAT EQUATION 6 2 TIME-DEPENDENT CONVECTION–DIFFUSION EQUATIONS 6 3 LINEAR PARABOLIC EQUATIONS OF SECOND ORDER 6 4 THE METHOD OF CHARACTERISTICS 6 5 THE TIME-DEPENDENT STOKES EQUATIONS 6 6 NONLINEAR PARABOLIC EQUATIONS OF SECOND ORDER 6 7 FINITE VOLUME METHODS 6 8 CONVERGENCE OF THE ADAPTIVE PROCESS III REFERENCES LIST OF SYMBOLS