numerical methods for nonlinear variable problems pot

[...]... Solution of Nonlinear Problems: Application to Nonlinear Problems in Fluid Dynamics 1 2 3 4 5 195 Introduction: Synopsis 195 Least-Squares Solution of Finite-Dimensional Systems of Equations 195 Least-Squares Solution of a Nonlinear Dirichlet Model Problem 198 Transonic Flow Calculations by Least-Squares and Finite Element Methods 211 Numerical Solution of the Navier-Stokes Equations for Incompressible... Elliptic Variational Inequality Methods to the Solution of Some Nonlinear Elliptic Equations 110 1 Introduction 2 Theoretical and Numerical Analysis of Some Mildly Nonlinear Elliptic Equations 3 A Subsonic Flow Problem 110 110 134 CHAPTER V Relaxation Methods and Applications 140 1 2 3 4 5 140 140 142 151 Generalities Some Basic Results of Convex Analysis Relaxation Methods for Convex Functionals: Finite-Dimensional... Relaxation Methods for Convex Functionals: Finite-Dimensional Case Block Relaxation Methods Constrained Minimization of Quadratic Functionals in Hilbert Spaces by Under and Over-Relaxation Methods: Application 6 Solution of Systems of Nonlinear Equations by Relaxation Methods 152 163 CHAPTER VI Decomposition-Coordination Methods by Augmented Lagrangian: Applications 166 1 2 3 4 5 6 7 166 168 170 171... Viscous Fluids by Least-Squares and Finite Element Methods 244 6 Further Comments on Chapter VII and Conclusion 318 Contents xiii APPENDIX I A Brief Introduction to Linear Variational Problems 321 1 2 3 4 321 321 326 Introduction A Family of Linear Variational Problems Internal Approximation of Problem (P) Application to the Solution of Elliptic Problems for Partial Differential Operators 5 Further Comments:... might think our approach is too mathematical for a book published in a collection oriented towards computational physics, we would like to say that many of the methods discussed here are used by engineers in industry for solving practical problems, and that, in our opinion, mastery of most of the tools of functional analysis used here is not too difficult for anyone with a reasonable background in applied... instead of problem (P 2 ) is that the bilinear form associated with (np) is the inner product of V which is symmetric Let us first assume that (TI") has a unique solution for all u e V and p > 0 For each p define the mapping fp: V -* V by fp(u) = w, where w is the unique solution of (n") We shall show that fp is a uniformly strict contraction mapping for suitably chosen p Let uu u2e V and wf = //«;),... Upwinding for Second-Order Problems with Large First- Order Terms 399 1 2 3 4 5 6 7 399 399 400 400 404 404 414 Introduction The Model Problem A Centered Finite Element Approximation A Finite Element Approximation with Upwinding On the Solution of the Linear System Obtained by Upwinding Numerical Experiments Concluding Comments APPENDIX III Some Complements on the Navier-Stokes Equations and Their Numerical. .. time the choice of the functional spaces used for the formulation and the solution of a given problem is not at all artificial, but is based on wellknown physical principles, such as energy conservation, the virtual work principle, and others From a computational point of view, a proper choice of the functional spaces used to formulate a problem will suggest, for example, what would be the "good" finite... V, • a(-,-): V x V -» U is a bilinear, continuous and V-elliptic form on V x V A bilinear form a{-, •) is said to be V-elliptic if there exists a positive constant a such that a(v, v) > oc\\v\\2, V v e V In general we do not assume a{-, •) to be symmetric, since in some applications nonsymmetric bilinear forms may occur naturally (see, for instance, Comincioli [1]) • L: V -> U continuous, linear functional,... Introduction Finite Element Approximation of the Boundary Condition u = g o n F i f g # 0 Some Comments On the Numerical Treatment of the Nonlinear Term (u • V)u Further Comments on the Boundary Conditions Decomposition Properties of the Continuous and Discrete Stokes Problems of Sec 4 Application to Their Numerical Solution 6 Further Comments 415 415 416 417 Some Illustrations from an Industrial Application . seemed at first glance, for several reasons: 1. The first version of Numerical Methods for Nonlinear Variational Problems was, in fact, part of a set of monographs on numerical mathe- matics. Szepessy, Stockholm, Sweden M.F. Wheeler, Austin, TX, USA Roland Glowinski Numerical Methods for Nonlinear Variational Problems With 82 Illustrations 123 Roland Glowinski University of Houston Dept and solution methods for some important problems in fluid dynamics are discussed, such as transonic flows for compressible inviscid fluids and the Navier-Stokes equations viii Preface for incompressible