MOVING-MESH FINITE ELEMENT METHOD WITH LOCAL REFINEMENT FOR PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS.

The authors 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. Details are also presented 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 problem-dependent numerical parameters.