Abstract
This chapter focuses on the finite element method and approximation theory. By a suitable choice of the trial functions ▪, the Galerkin equations for the coefficients ▪ turn out to be difference equations. It is to this combination of two fundamental techniques—to the fact that it is a variational method and at the same time takes the form of a difference equation—that the finite element method owes much of its success. The variational aspect is crucial to the formulation of the approximating scheme, allowing flexibility in the geometry and a physically intuitive derivation of accurate discrete analogues; in these respects, it is superior to more conventional difference equations. In the solution of these discrete analogs, it is the finite difference aspect that dominates. This chapter describes a systematic approach to the choice of such trial functions ▪.
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