论文标题
时间依赖的steklov-poincaré操作员和时空罗宾 - 罗宾蛋白分解热方程
Time-dependent Steklov--Poincaré operators and space-time Robin--Robin decomposition for the heat equation
论文作者
论文摘要
域分解方法是一组广泛使用的工具，用于平行偏微分方程求解器。对椭圆方程的收敛进行了很好的研究，但是在抛物线方程的情况下，几乎没有两个或多个维度的一般Lipschitz域的结果。因此，这项工作的目的是构建一个新框架，用于分析时空Lipschitz圆柱体中的热方程式的非重叠域分解方法。该框架是基于各种公式的，灵感来自最新的使用具有分数时间规律性的Sobolev空间对时空有限元素的研究。在此框架中，引入了与时间有关的Steklov-poincaré运营商，并证明了其基本特性。然后，我们得出了Dirichlet-Neumann，Neumann-Neumann- Neumann和Robin--Robin方法的界面解释，并表明这些方法已很好地定义。最后，我们证明了robin-robin方法的收敛性，并引入了具有更强收敛属性的修改方法。
Domain decomposition methods are a set of widely used tools for parallelization of partial differential equation solvers. Convergence is well studied for elliptic equations, but in the case of parabolic equations there are hardly any results for general Lipschitz domains in two or more dimensions. The aim of this work is therefore to construct a new framework for analyzing nonoverlapping domain decomposition methods for the heat equation in a space-time Lipschitz cylinder. The framework is based on a variational formulation, inspired by recent studies of space-time finite elements using Sobolev spaces with fractional time regularity. In this framework, the time-dependent Steklov--Poincaré operators are introduced and their essential properties are proven. We then derive the interface interpretations of the Dirichlet--Neumann, Neumann--Neumann and Robin--Robin methods and show that these methods are well defined. Finally, we prove convergence of the Robin--Robin method and introduce a modified method with stronger convergence properties.
