Multiscale Mixed Methods with Improved Accuracy: The Role of Oversampling and Smoothing

TL;DR

This paper introduces an improved multiscale mixed method with overlapping subdomains and smoothing steps, significantly boosting flux accuracy in heterogeneous subsurface flow simulations while maintaining computational efficiency.

Contribution

The study presents a novel multiscale mixed method using overlapping regions and smoothing to enhance flux accuracy without increasing computational cost.

Findings

Achieves two orders of magnitude improvement in flux accuracy.

Maintains computational cost close to traditional methods.

Effective for industry-relevant heterogeneous flow problems.

Abstract

Multiscale mixed methods based on non-overlapping domain decompositions can efficiently handle the solution of significant subsurface flow problems in very heterogeneous formations of interest to the industry, especially when implemented on multi-core supercomputers. Efficiency in obtaining numerical solutions is dictated by the choice of interface spaces that are selected: the smaller the dimension of these spaces, the better, in the sense that fewer multiscale basis functions need to be computed, and smaller interface linear systems need to be solved. Thus, in solving large computational problems, it is desirable to work with piecewise constant or linear polynomials for interface spaces. However, for these choices of interface spaces, it is well known that the flux accuracy is of the order of 10-1. This study is dedicated to advancing an efficient and accurate multiscale mixed method aimed at addressing industry-relevant problems. A distinctive feature of our approach involves subdomains with overlapping regions, a departure from conventional methods. We take advantage of the overlapping decomposition to introduce a computationally highly efficient smoothing step designed to rectify small-scale errors inherent in the multiscale solution. The effectiveness of the proposed solver, which maintains a computational cost very close to its predecessors, is demonstrated through a series of numerical studies. Notably, for scenarios involving modestly sized overlapping regions and employing just a few smoothing steps, a substantial enhancement of two orders of magnitude in flux accuracy is achieved with the new approach.