Domain decomposition methods for partial differential equations

科学和工程中的许多问题都被描述为偏微分方程组。例如，质量守恒、动量守恒和能量守恒可以用偏微分方程组表示。除了非常简单的情况外，没有解析解可用，我们经常依赖于偏微分方程组的数值解。这项拨款提案涉及一类称为区域分解方法的方法。在现代科学计算中，问题是如此之大，以至于没有一台计算机能够处理计算和内存需求。其想法是将域划分为较小的子域，并在并行计算机上将每个处理器分配给一个子域。我们并行求解每个子域上的偏微分方程组，然后将这些解拼接在一起形成全局解。这些都是迭代方法，这意味着通常情况下，计算出的解在无限次迭代后收敛到精确解。当然，在实践中，当满足规定的误差容限时，迭代停止。目标是一种最优收敛的数值格式，即以与离散化参数无关的速率收敛。另一个重要问题是如何定义沿每个内部边界的边界条件(S)。有一类方法称为优化Schwarz方法，它巧妙地在内边界条件中选择自由参数(S)来优化收敛速度。我们将研究二阶偏微分方程组以及更难的四阶偏微分方程组。后一类偏微分方程解的难度较大，因为它们比二阶偏微分方程组的病态程度高得多--对于四阶情形，传统迭代方法的收敛速度要慢得多。本程序将实现这些新的算法，并开发方法的收敛速度理论。如果成功，这些方法将使科学家和工程师能够解决更复杂的问题和/或以更高的分辨率解决问题。

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.