Adaptive mesh enrichment techniques are becoming recognized as a means of automating many of the decisions associated with the computer solution of partial differential equations. Towards this end, we describe an intelligent interface where partial differential equations and their data may be entered in a relatively natural mathematical language. Fortran subroutines for functions and Jacobians that are required by the numerical software are generated automatically with symbolic differentiation used to evaluate Jacobians. We further describe the use of this symbolic interface with an adaptive local mesh refinement finite element method for two-dimensional parabolic partial differential systems. Solutions are calculated using Galerkin's method with a piecewise bilinear polynomial basis in space and backward Euler integration in time. Estimates of the local discretization error, obtained by a p-refinement technique, are used to control a local h-refinement strategy where finer grids are recursively introduced in regions where a prescribed tolerance is exceeded. Fine grids at a given level of refinement may overlap each other and independent solutions are generated on each of them. A version of the Schwarz alternating principle is used to coordinate solutions between overlapping fine grids.