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

TL;DR

This paper extends the Multiscale Robin Coupled Method (MRCM) for subsurface flow by introducing oversampling in overlapping subdomains (MRCM-O) and a smoothing step (MRCM-OS). The approach leverages augmented subdomains $\hat{\Omega}_i$ and low-dimensional interface spaces $\Lambda^i_H$ to reduce resonance errors while preserving computational efficiency on multi-core HPC. Numerical experiments on analytic and SPE10-based heterogeneous problems show that oversampling yields notable accuracy gains, and combining oversampling with smoothing can achieve up to two orders of magnitude improvements in flux accuracy with minimal overhead, particularly for modest overlaps and $\alpha$ near 1. The results validate the method’s effectiveness with piecewise constant interface spaces and point toward extensions to informed spaces and three-dimensional applications, offering practical impact for industry-scale subsurface simulations.

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.

Figures (19)

• Figure 1: Decompositions of the computational domain $\Omega$: The non-overlapping subdomains are denoted by $\Omega_i$, and the corresponding oversampling regions are written as $\hat{\Omega}_i$. The three length scales that enter in the formulation of the MRCM with oversampling are also illustrated: $\hat{H} > H \geq h$.
• Figure 2: Partitions: non-overlapping (solid line) and oversampling (dotted line). In the coloring scheme oversampling subdomains sharing the same color do not have any common boundary points.
• Figure 3: Permeability field: Slice 40 of the SPE 10 project.
• Figure 4: $\alpha$ parameter study for the heterogeneous problem demonstrating significant improvement: Pressure relative error (left) and flux relative error (right).
• Figure 5: Multiscale solution for the heterogeneous problem presented in colored images for the Robin condition parameter $\alpha=1$: Original MRCM method (top) and MRCM-OS (bottom).

Theorems & Definitions (2)

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