Introduction to Finite Element Approximations of Variational Problems: (50 Pitman Monographs & Surveys in Pure & Applied Mathematics.)

The analytical solution of problems described by partial differential equations is known only in a few cases on special domains. Mostly it is thus necessary to use at least some numerical methods (statistical, topological etc) to get an approximation of the solution. One of the most useful numerical methods for solving partial differential equations seems to be the Galerkin method. The standard finite element method (FEM) is, roughly speaking, the Galerkin method with a special choice of basis functions. The FEM was first proposed in 1943 and ranks among the most efficient mumerical methods for solving problems of mathematical physics which are based on variational principles. The main aim of this book is to compress the standard theory of the FEM for elliptic problems into a form that can be understood by any reader who is familiar with basic knowledges in linear algebra and functional analysis. The second aim is to extend the standard material of books on the FEM to cover also some more special results obtained in this field. The authors consider among others: methods for increasing the accuracy in standard FE-approximations; methods for handling hyperbolic equations with nonhomogeneous boundary data; methods for solving Helmholtz's equation etc.

Table of Contents:
Variational formulation of second order elliptic problems - Sobolev spaces; finite element approximation; convergence of the finite element method; numerical integration; assembling the stiffness matrix; solving the discrete problem - the Jacobi method, the Gauss-Seidel method, Aiken's post-processing, the Choleski factorization; increasing the accuracy of finite element approximations - Richardson extrapolation; on the approximation of the solution to fourth order elliptic problems; on the approximationof the solution to a parabolic problem; on the approximation of the solution to hyperbolic problems; finite element approximations of systems with prescribed divergence and rotation - conforming approximation to the Stokes problem; FE- approximation of the solution to time-harmonic Maxwell's equations; on the numerical solution of Helmholtz's equation by different finite element methods - the Galerkin method; on the numerical solution of nonlinear problems; on the FE-approximation of an eigenvalue problem; on the FE-approximation of a bifurcation problem for the sine-Gordon type equation; on the solutin of problems with monotone operators - monotone operators in Hilbert spaces; axisymmetric elliptic problems - variational formulation in wieghted Sobolev spaces.