Recursive formulation and parallel implementation of multiscale mixed methods

Abstract

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.

The Multiscale Perturbation Method for Two-Phase Reservoir Flow Problems

“…This approximation follows from the fact that the computational cost of the interface problem is typically negligible when compared to the cost of computing BFs [18,19]. Furthermore, the downscaling step has essentially the same cost of computing one basis function at each subdomain (C DS ≈ C BF ).…”

“…We have recently established in [18,19] that a recursive formulation of the Multiscale Robin Coupled Method (MRCM) [16] shows excellent scalability (both weak and strong) for the solution of the pressure equation. These conclusions were reached by solving the pressure equation on state-of-the-art multi-core devices, for problems with a few billion variables, that are of interest to the oil industry.…”

“…These conclusions were reached by solving the pressure equation on state-of-the-art multi-core devices, for problems with a few billion variables, that are of interest to the oil industry. In [18,19] the solution of a second order elliptic equation is obtained in two steps. In a first step, for each subdomain of a decomposition of the domain of interest a set of multiscale basis functions (local boundary value problems of Robin type) has to be computed.…”

“…Then, a coarse interface problem defined on the skeleton of the domain decomposition needs to be solved. It has been shown in [18,19] that the time associated with the solution of the interface problem is essentially negligible, when compared to the time spent in solving the local boundary value problems that give the multiscale basis functions. Thus, a fair assessment of the cost of the solution of the pressure equation by a multiscale method can be made in terms of the number of multiscale basis functions that are computed.…”

A multiscale Robin-coupled implicit method for two-phase flows in high-contrast formations

“…They allow for the pressure and velocity fields to be computed on a coarse mesh (large scale), while detailed basis functions (locally defined for each subdomain) incorporate rock heterogeneity on a much finer grid (small scale) [2]. The local problems can be solved simultaneously in state-of-the-art parallel machines, making the simulation of huge problems feasible [3].…”