Discretizations on Non-Matching Grids and Domain Decomposition Methods
by
Raytcho D. Lazarov
Department of Mathematics, Texas A & M University
Coauthors: Joseph E. Pasciak (Texas A & M University), Panayot S. Vassilevski (LLNL)

Coupling or decomposing multidimensional problems from or to problems in smaller subdomains has become an important tool in the large scale numerical simulations.

This approach simplifies the meshing process, allows greater flexibility in using various approximation methods, and has strong potential for efficient parallel implementations and computations. In this talk we first introduce two hybrid formulations for second order elliptic problems, which are basis for the discretization methods that are used further. On nonmatching grids, we introduce coupling of finite volume approximations via mortars, coupling mixed finite elements via mortars, coupling mixed finite elements and finite volumes without mortars, and penalty formulation for standard Galerkin finite elements. In all cases we discuss the advantages and disadvantages of the method, outline the discrete spaces, and present the main steps in the stability and error analysis. Further, we introduce preconditioners for the problem on the composite nonmatching grid based on local subdomain solvers. Finally, we present model numerical experiments for second order elliptic problems related to groundwater and petroleum simulations and linear elastic deformations for nonmatching grids that are either given a priori or are obtained in the process of adaptive local grid refinement.

http://www.math.tamu.edu/~raytcho.lazarov/

Date received: February 24, 2000