Multiscale methods for second order elliptic equations based on non-overlapping domain decomposition schemes have great potential to take advantage of multi-core, state-of-the-art parallel computers. These methods typically involve solving local boundary value problems followed by the solution of a global interface problem. Known iterative procedures for the solution of the interface problem have typically slow convergence, increasing the overall cost of the multiscale solver. To overcome this problem we develop a scalable recursive solution method for such interface problem that replaces the global problem by a family of small interface systems associated with adjacent subdomains, in a hierarchy of nested subdomains. Then, we propose a novel parallel algorithm to implement our recursive formulation in multi-core devices using the Multiscale Robin Coupled Method by Guiraldello et al. (2018), that can be seen as a generalization of several multiscale mixed methods. Through several numerical studies we show that the new algorithm is very fast and exhibits excellent strong and weak scalability. We consider very large problems, that can have billions of discretization cells, motivated by the numerical simulation of subsurface flows.

中文翻译：

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多尺度混合方法的递归制定与并行实现

基于非重叠域分解方案的二阶椭圆方程的多尺度方法具有利用多核、最先进的并行计算机的巨大潜力。这些方法通常涉及解决局部边界值问题，然后解决全局接口问题。已知的用于求解接口问题的迭代程序通常收敛速度较慢，从而增加了多尺度求解器的总体成本。为了克服这个问题，我们为这种接口问题开发了一种可扩展的递归解决方案，该方法用嵌套子域层次结构中与相邻子域相关联的一系列小型接口系统来代替全局问题。然后，我们提出了一种新的并行算法，使用 Guiraldello 等人的多尺度罗宾耦合方法在多核设备中实现我们的递归公式。(2018)，这可以看作是几种多尺度混合方法的概括。通过几项数值研究，我们表明新算法速度非常快，并表现出出色的强弱可扩展性。我们考虑非常大的问题，这些问题可能有数十亿个离散单元，其动机是地下流动的数值模拟。