Fast Converging Parallel Offline-Online Iterative Multiscale Mixed Methods

TL;DR

This paper introduces two highly efficient iterative multiscale methods that significantly improve solution accuracy for complex permeability fields, achieving flux errors as low as 10^{-10} with rapid convergence.

Contribution

The paper develops two novel iterative multiscale methods based on online informed spaces, enhancing accuracy and efficiency over existing approaches.

Findings

Achieved flux error of 10^{-10} on SPE10 benchmark

Demonstrated rapid convergence of the proposed methods

Compared methods show improved efficiency over existing techniques

Abstract

In this work, we build upon the recently introduced Multiscale Robin Coupled Method with Oversampling and Smoothing (MRCM-OS) to develop two highly efficient iterative multiscale methods. The MRCM-OS methodology demonstrated the ability to achieve flux error magnitudes on the order of $10^{-4}$ in a challenging industry benchmark, namely the SPE10 permeability field. The two newly proposed iterative procedures, through the construction of online informed spaces, significantly enhance the solution accuracy, reaching flux error magnitudes of order $10^{-10}$ for a reduced number of steps. The proposed methods are based on the construction of online informed spaces, which are iteratively refined to improve solution accuracy. Following an initial offline stage, where known boundary conditions are applied to construct multiscale basis functions, the informed spaces are updated through iterative procedures that utilize boundary conditions defined by the most recently computed solution variables. Two distinct approaches are introduced, each leveraging this framework to deliver efficient and accurate iterative solutions. A series of numerical simulations, conducted on the SPE10 benchmark, demonstrates the very rapid convergence of the iterative solutions. These results highlight the computational efficiency and competitiveness of the two proposed methods, which are thoroughly compared to each other and to an existing multiscale iterative method from the literature.