The mathematical foundations of the finite element method with applications to partial differential equations proceedings of a symposium held at the University of Maryland, Baltimore, Maryland, June 26-30, 1972

The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations is a collection of papers presented at the 1972 Symposium by the same title, held at the University of Maryland, Baltimore County Campus. This symposium relates considerable numerical analysis involved in research in both theoretical and practical aspects of the finite element method. This text is organized into three parts encompassing 34 chapters. Part I focuses on the mathematical foundations of the finite element method, including papers on theory of approximation, variational principles, the problems of perturbations, and the eigenvalue problem. Part II covers a large number of important results of both a theoretical and a practical nature. This part discusses the piecewise analytic interpolation and approximation of triangulated polygons; the Patch test for convergence of finite elements; solutions for Dirichlet problems; variational crimes in the field; and superconvergence result for the approximate solution of the heat equation by a collocation method. Part III explores the many practical aspects of finite element method. This book will be of great value to mathematicians, engineers, and physicists. Content: Inside Front Cover, Page ii Front Matter, Page iii Copyright, Page iv CONTRIBUTORS, Pages ix-xi PREFACE, Page xiii FOREWORD, Pages 3-4, I. BABUŠKA, A.K. AZIZ CHAPTER 1 - PRELIMINARY REMARKS, Pages 5-14 CHAPTER 2 - THE FUNDAMENTAL NOTIONS, Pages 15-45 CHAPTER 3 - PROPERTIES OF SOLUTIONS OF ELLIPTIC BOUNDARY VALUE PROBLEMS, Pages 47-81 CHAPTER 4 - THEORY OF APPROXIMATION, Pages 83-110 CHAPTER 5 - VARIATIONAL PRINCIPLES, Pages 111-184 CHAPTER 6 - RATE OF CONVERGENCE OF THE FINITE ELEMENT METHOD, Pages 185-241 CHAPTER 7 - ONE PARAMETER FAMILIES OF VARIATIONAL PRINCIPLES, Pages 243-263 CHAPTER 8 - FINITE ELEMENT METHOD FOR NON-SMOOTH DOMAINS AND COEFFICIENTS, Pages 265-283 CHAPTER 9 - THE PROBLEMS OF PERTURBATIONS IN THE FINITE ELEMENT METHOD, Pages 285-301 CHAPTER 10 - THE EIGENVALUE PROBLEM, Pages 303-343 CHAPTER 11 - THE FINITE ELEMENT METHOD FOR TIME DEPENDENT PROBLEMS, Pages 345-359 PIECEWISE ANALYTIC INTERPOLATION AND APPROXIMATION IN TRIANGULATED POLYGONS, Pages 363-385, Garrett Birkhoff APPROXIMATION OF STEKLOV EIGENVALUES OF NON-SELFADJOINT SECOND ORDER ELLIPTIC OPERATORS, Pages 387-408, J.H. Bramble, J.E. Osborn THE COMBINED EFFECT OF CURVED BOUNDARIES AND NUMERICAL INTEGRATION IN ISOPARAMETRIC FINITE ELEMENT METHODS, Pages 409-474, P.G. Ciarlet, P.-A. Raviart A SUPERCONVERGENCE RESULT FOR THE APPROXIMATE SOLUTION OF THE HEAT EQUATION BY A COLLOCATION METHOD, Pages 475-490, Jim Douglas Jr. SOME L2 ERROR ESTIMATES FOR PARABOLIC GALERKIN METHODS, Pages 491-504, Todd Dupont COMPUTATIONAL ASPECTS OF THE FINITE ELEMENT METHOD, Pages 505-524, S.C. Eisenstat, M.H. Schultz EFFECTS OF QUADRATURE ERRORS IN FINITE ELEMENT APPROXIMATION OF STEADY STATE, EIGENVALUE AND PARABOLIC PROBLEMS, Pages 525-556, George J. Fix EXPERIENCE WITH THE PATCH TEST FOR CONVERGENCE OF FINITE ELEMENTS, Pages 557-587, Bruce M. Irons, Abdur Razzaque HIGHER ORDER SINGULARITIES FOR INTERFACE PROBLEMS, Pages 589-602, R.B. Kellogg ON DIRICHLET PROBLEMS USING SUBSPACES WITH NEARLY ZERO BOUNDARY CONDITIONS, Pages 603-627, J. Nitsche GENERALIZED CONJUGATE FUNCTIONS FOR MIXED FINITE ELEMENT APPROXIMATIONS OF BOUNDARY VALUE PROBLEMS, Pages 629-669, J.T. Oden FINITE ELEMENT FORMULATION BY VARIATIONAL PRINCIPLES WITH RELAXED CONTINUITY REQUIREMENTS, Pages 671-687, Theodore H.H. Pian VARIATIONAL CRIMES IN THE FINITE ELEMENT METHOD, Pages 689-710, Gilbert Strang SPLINE APPROXIMATION AND DIFFERENCE SCHEMES FOR THE HEAT EQUATION, Pages 711-746, Vidar Thomée THE EXTENSION AND APPLICATION OF SARD KERNEL THEOREMS TO COMPUTE FINITE ELEMENT ERROR BOUNDS, Pages 749-755, R.E. Barnhill, J.A. Gregory, J.R. Whiteman TWO TYPES OF PIECEWISE QUADRATIC SPACES AND THEIR ORDER OF ACCURACY FOR POISSON'S EQUATION, Pages 757-761, Alan E. Berger A METHOD OF GALERKIN TYPE ACHIEVING OPTIMUM L2 ACCURACY FOR FIRST ORDER HYPERBOLICS AND EQUATIONS OF SCHRÖDINGER TYPE, Pages 763-765, J.E. Dendy Jr. RICHARDSON EXTRAPOLATION FOR PARABOLIC GALERKIN METHODS, Pages 767-768, G. Fairweather, J.P. Johnson GEOMETRIC ASPECTS OF THE FINITE ELEMENT METHOD, Pages 769-783, William J. Gordon, Charles A. Hall THE USE OF INTERPOLATORY POLYNOMIALS FOR A FINITE ELEMENT SOLUTION OF THE MULTIGROUP DIFFUSION EQUATION, Pages 785-789, H.G. Kaper, G.K. Leaf, A.J. Lindeman A “LOCAL” BASIS OF GENERALIZED SPLINES OVER RIGHT TRIANGLES DETERMINED FROM A NONUNIFORM PARTITIONING OF THE PLANE, Page 791, Lois Mansfield LEAST SQUARE POLYNOMIAL SPLINE APPROXIMATION, Pages 793-796, Paul D. Patent SUBSPACES WITH ACCURATELY INTERPOLATED BOUNDARY CONDITIONS, Page 797, Ridgway Scott Edited By A. K. Aziz. Sponsored By The Division Of Mathematics, University Of Maryland, Baltimore County Campus, And The U.s. Office Of Naval Research. Includes Bibliographical References.