Methods Based on More General Approximations of the Elliptic Problem

Abstract

In our above discussion of finite element approximation of the parabolic problem, the discretization in space was based on using a family of finitedimensional spaces S _h ⊂ H¹₀ = H¹₀ (Ω), such that, for some r ≥ 2, the approximation property (1.10) holds. The most natural example of such a family in a plane domain Ω is to take for S _h the continuous functions which reduce to polynomials of degree at most r−1 on the triangles τ of a triangulation T _h of Ω of the type described in the beginning of Chapter 1, and which vanish on ∂Ω However, for r > 2 and in the case of a domain with smooth boundary, it is not possible, in general, to satisfy the homogeneous boundary conditions exactly for this choice. This difficulty occurs, of course, already for the elliptic problem, and several methods have been suggested to deal with it. In this chapter we shall consider, as a typical example, a method which was proposed by Nitsche for this purpose.