Adaptive finite element techniques provide a reliable, robust, and efficient means of solving the complex partial differential equations that arise in practice. Likewise, parallel computation is becoming indispensable in the solution of modern large-scale scientific engineering computational problems. We propose to investigate parallel and adaptive procedures for solving vector systems of three-dimensional time-dependent parabolic equations. With a goal of developing the most efficient techniques, we will emphasize adaptive procedures that include both mesh modification (refinement and coarsening) and variation of the order of the finite element procedure. In a similar spirit, we will concentrate on massive parallelism using SIMD and MIMD architectures. Resulting software will be capable of solving problems on arbitrary three-dimensional problem domains using a finite octree automatic mesh generation procedure. This tree structure, which manages data associated with mesh generation and adaptive mesh refinement, with further be used to schedule parallel processors so that loading is balanced and to map spatial regions to various parallel architectures. Likewise, we will explore procedures that utilize the a posteriori error estimates, furnished with the adaptive software, to estimate the work remaining in different portions of the problem domain and schedule processors accordingly.
