Parallel Adaptive Finite Element Methods for Parabolic Partial Differential Equations

自适应有限元技术为解决实际中出现的复杂偏微分方程提供了一种可靠、稳健和有效的方法。同样，并行计算在解决现代大规模科学工程计算问题中也变得不可或缺。我们建议研究求解三维时变抛物方程矢量系统的并行和自适应程序。以开发最有效的技术为目标，我们将强调自适应程序，包括网格修改（细化和粗化）和有限元程序顺序的变化。本着类似的精神，我们将关注使用SIMD和MIMD架构的大规模并行性。由此产生的软件将能够解决任意三维问题域上的问题，使用有限八叉树自动网格生成程序。这种树状结构管理与网格生成和自适应网格细化相关的数据，可以进一步用于调度并行处理器，以便平衡负载，并将空间区域映射到各种并行架构。同样地，我们将探索利用后验误差估计的过程，配备自适应软件，以估计问题域不同部分的剩余工作并相应地调度处理器。

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.