A moving-mesh finite element method with local refinement for parabolic partial differential equations

Abstract

We discuss a moving-mesh finite element method for solving initial boundary value problems for vector systems of partial differential equations in one space dimension and time. The system is discretized using piecewise linear finite element approximations in space and a backward difference code for stiff ordinary differential systems in time. A spatial-error estimation is calculated using piecewise quadratic approximations that use the superconvergence properties of parabolic systems to gain computational efficiency. The spatial-error estimate is used to move and locally refine the finite element mesh in order to equidistribute a measure of the total spatial error and to satisfy a prescribed error tolerance. Ordinary differential equations for the spatial-error estimate and the mesh motion are integrated in time using the same backward difference software that is used to determine the numerical solution of the partial differential system. We present several details of an algorithm that may be used to develop a general-purpose finite element code for one-dimensional parabolic partial differential systems. The algorithm combines mesh motion and local refinement in a relatively efficient manner and attempts to eliminate problemdependent numerical parameters. A variety of examples that motivate our mesh-moving strategy and illustrate the performance of our algorithm are presented.