大规模非线性椭圆问题的并行外推瀑布式多网格法研究

Study of nonlinear elliptic problems comes from a large number of problems that arise in hydromechanics, Engineering Technology and the economic and social system, and has become today's one of the most interesting, but difficult problem. How to design high-performance (high-precision, high-efficiency) algorithms to solve these large-scale nonlinear elliptic problems is a core problem of modern scientific computing. This project will adopt a weak, but suitable for the overall measure discrete L2 norm to study the error asymptotic expansions of finite element with interpolated coefficient with block uniform grid for solving nonlinear elliptic problems. Thus we can construct high precision extrapolation formulas, which approximate finite element solution and the true solution, then propose an extrapolation cascadic multigrid method (EXCMG) based on interpolation coefficient finite element, and analyze the convergence of the method theoretically. For a large-scale three-dimensional problem, parallel EXCMG algorithms for solving nonlinear elliptic problems can be presented based on the OpenMP and MPI, respectively. Finally, EXCMG algorithm for solving large-scale three-dimensional nonlinear elliptic problems can be implemented on a machine with high parallel computation capability, and a large number of typical examples can be carried out to test the feasibility of the method. The expected results of the project will not only enrich the error theory of nonlinear elliptic problems , but also expand the range of applications of the EXCMG algorithm, and solve a large number of the problems encountered in practice, which has important theoretical significance and practical value.

非线性椭圆问题的研究来源于流体力学、工程技术及经济社会系统中的大量问题，现已成为当今最有趣，也最困难的大课题。如何高性能（高精度、高效率）求解这些大规模的非线性椭圆问题是现代科学计算的核心问题。本项目拟采用一种较弱的、却适合整体度量的离散L2范数，在分块均匀网格下研究插值系数有限元法求解非线性椭圆问题时有限元解的误差渐近展开式。由此构造逼近有限元解和真解的高精度外推公式，进而提出基于插值系数有限元的外推瀑布式多网格法(EXCMG)，并从理论上分析方法的收敛性。针对大规模三维问题，基于OpenMP和MPI分别提出求解非线性椭圆问题的并行EXCMG算法。最终在并行机上实现求解大规模三维非线性椭圆问题的EXCMG算法，并通过大量典型算例验证方法的可行性。本项目的预期成果不仅能丰富非线性椭圆问题的误差分析理论，又能扩大EXCMG法的应用范围，解决实际中碰到的大量问题，具有重要的理论意义和实用价值。

非线性椭圆问题的研究来源于流体力学、工程技术及经济社会系统中的大量问题，现已成为当今最有趣，也最困难的大课题。本项目主要讨论了求解线性和非线性问题的外推瀑布式多网格法(EXCMG)。得到了求解线性椭圆问题EXCMG算法的最佳收敛性和超最优性结果；并证明了非线性椭圆问题多水平线线性化EXCMG的最佳收敛性估计；还推导了凸角域上的椭圆问题，基于分块几乎均匀网格下三角形线性元在H1范数意义下的整体超收敛性和离散L2范数意义下的外推公式；另外，还讨论了EXCMG对求解3维椭圆问题的高效性。作为算法的推广，借鉴椭圆问题的研究经验，还提出了求解线性抛物问题的两种高性能新算法：直接外推瀑布多网格法(DEXCMG)和时间外推算法(TEA)，及求解变系数抛物问题的矩阵外推算法(MTEA)。本项目的成果不仅丰富了非线性椭圆问题的误差分析理论，又能扩大了高性能算法的应用范围，具有重要的理论意义和实用价值。