Algorithms and data structures for adaptive multigrid elliptic solvers

Abstract

With the advent of multigrid iteration, the large linear systems arising in numerical treatment of elliptic boundary value problems can be solved quickly and reliably. This frees the researcher to focus on the other issues involved in numerical solution of elliptic problems: adaptive refinement, error estimation and control, and grid generation. Progress is being made on each of these issues and the technology now seems almost at hand to put together general purpose elliptic software having reliability and efficiency comparable to that of library software for ordinary differential equations. This paper looks at the components required in such general elliptic solvers and suggests new approaches to some of the issues involved. One of these issues is adaptive refinement and the complicated data structures required to support it. These data structures must be carefully tuned, especially in three dimensions where the time and storage requirements of algorithms are crucial. Another major issue is grid generation. The options available seem to be curvilinear fitted grids, constructed on interactive graphics systems, and unfitted Cartesian grids, which can be constructed automatically. On several grounds, including storage requirements, the second option seems preferrable for the well behaved scalar elliptic problems considered here. A variety of techniques for treatment of boundary conditions on such grids have been described previously and are reviewed here. A new approach, which may overcome some of the difficulties encountered with previous approaches, is also presented.