THE MATHEMATICS OF CIRCUIT ANALYSIS

By LINDA WILSON

CONTENTS I ELLIPTIC PROBLEMS ONE-DIMENSIONAL PROBLEM MODEL PROBLEM A CLASS OF DG METHODS EXISTENCE AND UNIQUENESS OF THE DG SOLUTION LINEAR SYSTEM COMPUTING THE MATRIX A COMPUTING THE RIGHT-HAND SIDE B IMPOSING BOUNDARY CONDITIONS STRONGLY CONVERGENCE OF THE DG METHOD NUMERICAL EXPERIMENTS BIBLIOGRAPHICAL REMARKS EXERCISES HIGHER DIMENSIONAL PROBLEM PRELIMINARIES VECTOR NOTATION SOBOLEV SPACES TRACE THEOREMS APPROXIMATION PROPERTIES GREEN’S THEOREM CAUCHY–SCHWARZ’S AND YOUNG’S INEQUALITIES WEAK SOLUTION NUMERICAL SOLUTION BROKEN SOBOLEV SPACES JUMPS AND AVERAGES VARIATIONAL FORMULATION CONSISTENCY FINITE ELEMENT SPACES REFERENCE ELEMENTS VERSUS PHYSICAL ELEMENTS BASIS FUNCTIONS NUMERICAL QUADRATURE DG SCHEME PROPERTIES COERCIVITY OF BILINEAR FORMS CONTINUITY OF BILINEAR FORM LOCAL MASS CONSERVATION EXISTENCE AND UNIQUENESS OF DG SOLUTION ERROR ANALYSIS ERROR ESTIMATES IN THE ENERGY NORM ERROR ESTIMATES IN THE L2 NORM IMPLEMENTING THE DG METHOD DATA STRUCTURE LOCAL MATRICES AND RIGHT-HAND SIDES GLOBAL MATRIX AND RIGHT-HAND SIDE SMOOTH SOLUTION SINGULAR SOLUTION CONDITION NUMBER THE LOCAL DISCONTINUOUS GALERKIN METHOD DE?NITION OF THE MIXED DG METHOD EXISTENCE AND UNIQUENESS OF THE SOLUTION A PRIORI ERROR ESTIMATES DG VERSUS CLASSICAL ?NITE ELEMENT METHOD II PARABOLIC PROBLEMS PURELY PARABOLIC PROBLEMS FUNCTIONAL SPACES GRONWALL’S INEQUALITIES TAYLOR’S EXPANSIONS POINCARɒS INEQUALITIES INVERSE INEQUALITIES SEMIDISCRETE FORMULATION A PRIORI BOUNDS ERROR ESTIMATES FULLY DISCRETE FORMULATION BACKWARD EULER DISCRETIZATION FORWARD EULER DISCRETIZATION CRANK–NICOLSON DISCRETIZATION RUNGE–KUTTA DISCRETIZATION DG IN TIME DISCRETIZATION IMPLEMENTATION PARABOLIC PROBLEMS WITH CONVECTION EXISTENCE AND UNIQUENESS OF SOLUTION OVERSHOOT AND UNDERSHOOT SLOPE LIMITERS AN IMPROVED DG METHOD III APPLICATIONS LINEAR ELASTICITY STRAIN AND STRESS TENSORS KORN’S INEQUALITIES LOCAL EQUILIBRIUM COERCIVITY STOKES ?OW BARYCENTRIC COORDINATES AN APPROXIMATION OPERATOR OF DEGREE ONE AN APPROXIMATION OPERATOR OF HIGHER DEGREE LOCAL L2 PROJECTION GENERAL INF-SUP CONDITION MODEL PROBLEM AND WEAK SOLUTION DISCRETE INF-SUP CONDITION NUMERICAL RESULTS NAVIER–STOKES ?OW SOBOLEV IMBEDDING HÖLDER’S INEQUALITY BROUWER’S ?XED POINT THEOREM DG DISCRETIZATION NONLINEAR CONVECTIVE TERM SCHEME EXISTENCE OF DISCRETE VELOCITY EXISTENCE OF DISCRETE PRESSURE UNIQUENESS EFFECTS OF PENALTY SIZE STEP CHANNEL PROBLEM FLOW IN POROUS MEDIA TWO-PHASE ?OW A SEQUENTIAL APPROACH A COUPLED APPROACH NUMERICAL EXAMPLES MISCIBLE DISPLACEMENT A FULLY DISCRETE APPROACH