Piecewise Polynomial Approximation in 2D

Abstract

In this chapter we extend the concept of piecewise polynomial approximation to two dimensions. As before, the basic idea is to construct spaces of piecewise polynomial functions that are easy to represent in a computer and to show that they can be used to approximate more general functions. A difficulty with the construction of piecewise polynomials in higher dimension is that first the underlying domain must be partitioned into elements, such as triangles, which may be a nontrivial task if the domain has complex shape. We present a methodology for building representations of piecewise polynomials on triangulations that is efficient and suitable for computer implementation and study the approximation properties of these spaces. 3.1 Meshes 3.1.1 Triangulations Let ˝ R 2 be a bounded two-dimensional domain with smooth or polygonal boundary @˝. A triangulation, or mesh, K of ˝ is a set fKg of triangles K, such that ˝ D [ K2K K, and such that the intersection of two triangles is either an edge, a corner, or empty. No triangle corner is allowed to be hanging, that is, lie on an edge of another triangle. The corners of the triangles are called the nodes. Figure 3.1 shows a triangle mesh of the Greek letter. The set of triangle edges E is denoted by E D fEg. We distinguish between edges lying within the domain ˝ and edges lying on the boundary @˝. The former belongs to the set of interior edges E I , and the latter to the set of boundary edges E B , respectively. To measure the size of a triangle K we introduce the local mesh size h K , defined as the length of the longest edge in K. See Fig. 3.5. Moreover, to measure the quality