Locally Refined Tessellations

In all of the previous chapters, we assumed that the problem domain Ω is a tessellation. Three of the essential features in the definition (4.5.1) of a tessellation are that the closure of a tessellation domain is the union of the closures of its disjoint elements, the boundaries of the elements are a union of element faces, and the element faces either lie in the boundary of Ω or shared with exactly one adjacent element. The purpose of the last clause is to prevent hanging nodes, which do not coincide with vertices of the adjacent element. Hanging nodes lead to auxiliary constraints in conforming finite element linear systems, in which the degrees of freedom in fine adjacent element faces are constrained through continuity conditions by degrees of freedom in a course element face.

There are no hanging nodes in the refinement of one-dimensional intervals. Hanging nodes can be avoided by bisection of triangles, as described by Bank [36], and bisection of tetrahedra, as described by Arnold et al. [19]. However, substantial care must be taken in bisection to preserve a locally quasi-uniform mesh. Quadrilaterals, wedges and hexahedra can also be bisected, but refinement of tessellations involving these shapes may be non-local if hanging nodes are avoided. For example, a rectangular tessellation in which one element is bisected horizontally will require that all elements in that row be bisected horizontally in order to avoid hanging nodes. These three shapes are quite naturally subdivided into 2d elements with shape regularity (ratio of radius of circumscribed sphere to radius of inscribed sphere) the same as or better than the parent.