Elliptic Differential Equations: Theory and Numerical Treatment

Description

This book simultaneously presents the theory and the numerical treatment of elliptic boundary value problems, since an understanding of the theory is necessary for the numerical analysis of the discretisation. It first discusses the Laplace equation and its finite difference discretisation before addressing the general linear differential equation of second order. The variational formulation together with the necessary background from functional analysis provides the basis for the Galerkin and finite-element methods, which are explored in detail. A more advanced chapter leads the reader to the theory of regularity. Individual chapters are devoted to singularly perturbed as well as to elliptic eigenvalue problems. The book also presents the Stokes problem and its discretisation as an example of a saddle-point problem taking into account its relevance to applications in fluid dynamics. Front Matter....Pages i-xiv Partial Differential Equations and Their Classification Into Types....Pages 1-12 The Potential Equation....Pages 13-28 The Poisson Equation....Pages 29-42 Difference Methods for the Poisson Equation....Pages 43-91 General Boundary-Value Problems....Pages 93-118 Tools from Functional Analysis....Pages 119-158 Variational Formulation....Pages 159-180 The Finite-Element Method....Pages 181-262 Regularity....Pages 263-310 Special Differential Equations....Pages 311-327 Elliptic Eigenvalue Problems....Pages 329-354 Stokes Equations....Pages 355-380 Back Matter....Pages 381-455