Variational Methods for Numerical Solution of Boundary-Value Problems for Differential Equations. Finite Element Method. Boundary Element Method

Abstract

Among numerical methods of solution of linear boundary-value problems for ordinary differential equations and elliptic partial differential equations, variational methods play an important role. Theoretical results for these methods have been obtained mostly by means of functional analysis. Thus the reader is recommended to have a look at Definitions 22.4.6 and 22.4.7 at first, concerning the concept of the Hilbert space, and at Remark 22.4.10 on generalized derivatives and the Sobolev space. This space, currently used in variational methods, is mostly denoted by H ^k(Ω) or W ^k ₂ (Ω). In this chapter, the former symbol is used. If no misunderstanding can occur we use only the symbol H ^k instead of H ^k(Ω). Further, the reader is recommended to notice Theorem 22.6.9 on the minimum of the so-called functional of energy, Remark 22.6.10 on the energy space H _A and the generalized solution to operator equations of the form Au f, and §§ 18.8 and 18.9 about generalized and weak solutions of differential equations. These concepts play a fundamental role in variational methods.