Domain decomposition methods for partial differential equations

偏微分方程的域分解方法

• 批准号: 250303-2011
• 负责人: Lui, ShiuHong(Shaun)
• 金额: $ 0.73万
• 依托单位: University of Manitoba
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2014
• 资助国家: 加拿大
• 起止时间: 2014-01-01 至 2015-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=547584
• 关键词: Domain; decomposition; methods; partial; differential

Many problems in science and engineering are posed as partial differential equations (PDEs). For instance, conservations of mass, momentum and energy can be expressed as PDEs. Except for very simple cases, no analytic solution is available and we often rely on a numerical solution of the PDEs. This grant proposal deals with a class of methods called domain decomposition methods. In modern scientific computing, the problems are so large that no single computer can handle the computational and memory demands. The idea is to split up the domain into smaller subdomains and on a parallel computer, assign each processor to a subdomain. We solve the PDE on each subdomain in parallel and then stitch together these solutions to form the global solution. These are iterative methods, meaning that typically, the computed solution converges to the exact solution after infinitely many iterations. Of course in practice, the iteration is stopped when a prescribed error tolerance is satisfied. The goal is a numerical scheme which converges optimally, that is, at a rate which is independent of the discretization parameter. Another important issue is how to define the boundary condition(s) along each interior boundary. There is a subclass of methods, called Optimized Schwarz methods, which cleverly chooses free parameter(s) in the interior boundary conditions to optimize the convergence rate. We shall work on second-order PDEs as well as more difficult fourth-order PDEs. The latter PDEs are more difficult because they are far more ill-conditioned than second-order PDEs - the rate of convergence of traditional iterative methods is far slower for the fourth-order case. This program will implement these new algorithms as well as develop a theory of convergence rate of the methods. If successful, these methods will allow scientists and engineers to solve more complex problems and/or solve problems with a higher resolution.

科学和工程中的许多问题都是作为部分微分方程（PDE）提出的。 例如，可以将质量，动量和能量的保护节表示为PDE。 除了非常简单的情况外，没有分析解决方案可用，我们通常依靠PDE的数值解决方案。 该赠款提案涉及一类称为域分解方法的方法。 在现代科学计算中，问题是如此之大，以至于没有一家计算机能够处理计算和内存需求。 这个想法是将域分为较小的子域，并在平行计算机上，将每个处理器分配给子域。我们并行在每个子域上求解PDE，然后将这些解决方案缝合在一起以形成全局溶液。 这些是迭代方法，这意味着通常，计算的解决方案在无限的许多迭代后会收敛到精确的解决方案。 当然，实际上，当满足规定的错误公差时，迭代就会停止。 目标是一个数值方案，该方案以与离散参数无关的速率达到最佳收敛。 另一个重要的问题是如何沿每个内部边界定义边界条件。 有一个方法的亚类，称为优化的Schwarz方法，它巧妙地在内部边界条件下巧妙地选择了自由参数以优化收敛速率。 我们将在二阶PDE以及更困难的四阶PDE上工作。 后者的PDE更加困难，因为它们的条件比二阶PDE差得多 - 对于四阶情况，传统迭代方法的收敛速度要慢得多。该程序将实施这些新算法，并发展方法的收敛速率理论。 如果成功，这些方法将使科学家和工程师可以通过更高的分辨率解决更复杂的问题和/或解决问题。