High-order adaptive methods for parabolic systems

Abstract

We consider the adaptive solution of parabolic partial differential systems in one and two space dimensions by finite element procedures that automatically refine and coarsen computational meshes, vary the degree of the piecewise polynomial basis and, in one dimension, move the computational mesh. Two-dimensional meshes of triangular, quadrilateral, or a mixture of triangular and quadrilateral elements are generated using a finite quadtree procedure that is also used for data management. A posteriori estimates, used to control adaptive enrichment, are generated from the hierarchical polynomial basis. Temporal integration, within a method-of-lines framework, uses either backward difference methods or a variant of the singly implicit Runge-Kutta (SIRK) methods. A high-level user interface facilitates use of the adaptive software.