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.

翻译：基于非重叠区域分解的多尺度混合方法，能够高效处理工业界关注的强非均质性地层中重大地下流动问题的求解，尤其是在多核超级计算机上实现时尤为有效。数值求解的效率取决于所选界面空间的维度：这些空间维度越小越好，因为这意味着需要计算的多尺度基函数数量更少，且需求解的界面线性系统规模更小。因此，在求解大型计算问题时，采用分段常数或线性多项式作为界面空间是可取的。然而，众所周知，这些界面空间选择下的流量精度仅为10^{-1}量级。本研究致力于改进一种高效且精确的多尺度混合方法，以解决工业相关难题。我们方法的独特之处在于采用具有重叠区域的子域，这与传统方法不同。我们利用重叠分解引入一种计算高效的平滑步骤，旨在修正多尺度解中固有的小尺度误差。通过一系列数值研究，证明了所提求解器的有效性，其计算成本与先前方法几乎持平。值得注意的是，在采用适度大小的重叠区域并仅执行少量平滑步骤的情况下，新方法实现了流量精度两个量级的显著提升。