In this work we develop a novel domain splitting strategy for the solution of partial differential equations. Focusing on a uniform discretization of the $d$-dimensional advection-diffusion equation, our proposal is a two-level algorithm that merges the solutions obtained from the discretization of the equation over highly anisotropic submeshes to compute an initial approximation of the fine solution. The algorithm then iteratively refines the initial guess by leveraging the structure of the residual. Performing costly calculations on anisotropic submeshes enable us to reduce the dimensionality of the problem by one, and the merging process, which involves the computation of solutions over disjoint domains, allows for parallel implementation.

翻译：本文提出了一种新颖的区域分割策略，用于求解偏微分方程。以$d$维对流-扩散方程的均匀离散化为例，我们设计了一种两层级算法：该算法通过融合在高度各向异性子网格上对方程进行离散化所获得的解，计算精细解的初始近似值。随后，算法利用残差的结构性特征对初始猜测进行迭代精化。在高度各向异性子网格上执行高计算成本的操作，使得问题维度降低一维；而融合过程涉及对互不相交区域上解的计算，从而支持并行实现。