Non-Matching Grid Discretizations

A non-matching grid is a collection of overlapping or non-overlapping grids, with associated subdomains that cover a domain, where the grids are obtained by the independent triangulation of the subdomains, without requirement to match with the grids adjacent to it, see Fig. 11.1. In this chapter, we describe several methods for the global discretization of a self adjoint and coercive elliptic equation on a non-matching grid:

• Mortar element discretization of an elliptic equation.

• Chimera (composite grid or Schwarz) discretization of an elliptic equation.

• Alternative non-matching grid discretizations of an elliptic equation.

Each non-matching grid discretization is based on a hybrid formulation of the underlying elliptic equation on its associated subdomain decomposition. The mortar element method, for instance, is formulated for a non-overlapping non-matching grid, and employs a Lagrange multiplier hybrid formulation of the elliptic equation, which enforces weak matching of the solution across adjacent subdomains [MA4, BE18, BE23, BE6, BE4, WO, WO4, WO5, KI], while the Chimera discretization is a finite difference discretization on an overlapping grid that enforces strong matching of the solution across adjacent grids, using a Schwarz formulation [ST, ST6, GR16, HE9, HE10, GO7, CA17].