A Local Refinement Finite-Element Method for Two-Dimensional Parabolic Systems

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Abstract
We discuss an adaptive local refinement finite-element procedure for solving initial boundary value problems for vector systems of parabolic partial differential equations on two-dimensional rectangular regions. The differential equations are discretized in space using piecewise bilinear finite-element approximations. An estimate of the spatial discretization error of the solution is obtained using piecewise cubic polynomials that employ nodal superconvergence to gain computational efficiency. The resulting system of ordinary differential equations for the finite-element solution and error estimate are integrated in time using existing software for stiff differential systems. The spatial error estimate is used to locally refine the finite-element mesh in order to satisfy a prescribed error tolerance. We discuss several aspects of the refinement algorithm and the dynamic tree data structure that is used to store the mesh, solution, and error estimate. A code that is based on our methods is applied to sever...