Nonlinear Partial Differential Equations in Engineering and Applied Science

Robert L. Sternberg, Anthony J. Kalinowski, John S. Papadakis

• Publisher: CRC Press
• Year: 2017
• Print ISBN: 9780824769963
• eBook ISBN: 9781351428057
• Topic: Mathematics
• Subtopic: Differential Equations
• Index: Mathematics

In this volume are twenty-eight papers from the Conference on Nonlinear Partial Differential Equationsin Engineering and Applied Science, sponsored by the Office of Naval Research and held at the Universityof Rhode Island in June, 1979. Included are contributions from an international group of distinguishedmathematicians, scientists, and engineers coming from a wide variety of disciplines and having a commoninterest in the application of mathematics, particularly nonlinear partial differential equations, to realworld problems.The subject matter ranges from almost purely mathematical topics in numerical analysis and bifurcationtheory to a host of practical applications that involve nonlinear partial differential equations, suchas fluid dynamics, nonlinear waves, elasticity, viscoelasticity, hyperelasticity, solitons, metallurgy, shocklessairfoil design, quantum fields, and Darcy's law on flows in porous media.Non/inear Partial Differential Equations in Engineering and Applied Science focuses on a variety oftopics of specialized, contemporary concern to mathematicians, physical and biological scientists, andengineers who work with phenomena that can be described by nonlinear partial differential equations.

ON THE STRUCTURE OF THE ERRORS MADE IN MIXED FINITE ELEMENT METHODS

George J. Fix

Carnegie-Mellon University Pittsburgh, Pennsylvania

§1. INTRODUCTION

Mixed finite element methods differ from standard finite element schemes in their use of stationary variational principles, i.e., principles which characterize the solution of the problem to be approximated as a stationary point rather than as a strict minimum or maximum. One consequence of this property is the possibility of instabilities in the approximation, and current research has emphasized the establishment of conditions which insure both convergence and stability. From a practical standpoint it is important that these conditions be as sharp as possible so that the user not be forced into needless complications to obtain stable and accurate approximations.

In a ...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• CONTRIBUTORS
• INTRODUCTION
• ON THE EXISTENCE OF HYDROSTATIC PRESSURE IN REGULAR FINITE DEFORMATIONS OF INCOMPRESSIBLE HYPERELASTIC SOLIDS
• RIGIDITY AND NONLINEAR ELASTICITY
• ELECTRICAL INTERACTIONS BETWEEN PARALLEL NERVE FIBERS: A MODEL AND SOME RESULTS
• UNSTABLE VISCOELASTIC FLUID FLOWS
• ISOLAS AND PERTURBED BIFURCATION THEORY
• ON THE STRUCTURE OF THE ERRORS MADE IN MIXED FINITE ELEMENT METHODS
• SOLUTIONS OF THE KORTEWEG-DEVRIES EQUATION
• A THREE-PHASE FREE BOUNDARY PROBLEM ARISING IN ALLOY SOLIDIFICATION
• SPATIAL DECAY FOR THE NAVIER-STOKES EQUATIONS
• A RIEMANNIAN METRIC FOR PARTIAL DIFFERENTIAL EQUATIONS
• SHOCKLESS AIRFOILS FOR WINGS, COMPRESSORS, AND TURBINES
• SOME SINGULARLY PERTURBED NONLINEAR BOUNDARY VALUE PROBLEMS OF ELLIPTIC TYPE
• THREE-DIMENSIONAL NONLINEAR EVOLUTION OF WATER WAVES
• A COMPARATIVE STUDY OF NUMERICAL SCHEMES FOR THE SOLUTION OF TWO-DIMENSIONAL ADVECTION FLOWS
• DECAY OF SOLUTIONS TO THE KORTEWEG DE VRIES-BURGER EQUATION AND TO PARABOLIC CONSERVATION LAWS
• SMALL AMPLITUDE STABLE WAVETRAINS IN REACTION-DIFFUSION SYSTEMS
• A SINGULARITY THEORY APPROACH TO STEADY-STATE BIFURCATION THEORY
• SYMMETRY PROPERTIES AND ISOLATED SINGULARITIES OF POSITIVE SOLUTIONS OF NONLINEAR ELLIPTIC EQUATIONS
• SOME RESULTS ON A REACTION-DIFFUSION SYSTEM
• DISCRETIZING THE SINE-GORDON EQUATION: GEOMETRICAL AND GROUP THEORETICAL ASPECTS
• ADVANCES IN THE NUMERICAL COMPUTATION OF CAPILLARY-GRAVITY WAVES
• CLASSICAL AND QUANTUM DYNAMICS OF INTEGRABLE NONLINEAR FIELD THEORIES
• FINITE-DIFFERENCE SCHEME FOR STATIONARY NAVIER-STOKES EQUATIONS WITH VARIABLE COEFFICIENTS
• MULTIPHASE SIMILARITY SOLUTIONS OF INTEGRABLE EVOLUTION EQUATIONS
• SOLUTIONS OF AN INTERNAL WAVE EQUATION DESCRIBING A STRATIFIED FLUID WITH FINITE DEPTH
• JUMP PHENOMENA
• DARCY’S LAW FOR FLOW IN POROUS MEDIA AND THE TWO-SPACE METHOD
• DECOMPOSITION OF VARIATIONAL PROBLEMS AND APPLICATIONS IN FINITE ELASTICITY