Fast Converging Parallel Offline-Online Iterative Multiscale Mixed Methods

TL;DR

This work extends the Multiscale Robin Coupled Method with Oversampling and Smoothing (MRCM-OS) by introducing two offline-online staged iterative strategies, the Reduced Method (RM) and Extended Method (EM), to generate online informed spaces that dramatically improve solution accuracy with minimal iterations. By coupling MRCM-OS with online MBF enrichment and adaptive boundary conditions, the authors achieve flux errors as small as $10^{-10}$ in about ten iterations on challenging SPE10 permeability fields, outperforming a representative existing method in convergence speed. Key insights include the beneficial roles of oversampling and smoothing, and the existence of an optimal Robin parameter $\\alpha$ (found to be $10$ in their tests) that minimizes iteration counts. The framework shows strong potential for scalable, parallelized subsurface flow simulations and may be extended to 3D preconditioning and related multiscale problems.

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.

Figures (11)

• 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: Permeability field: Slice 42 (left) and slice 84 (right) of the SPE 10 project.
• Figure 3: $\alpha$ parameter study for slice 42: iteration times for pressure (left) and flux (right)
• Figure 4: $\alpha$ parameter study for slice 84: iteration times for pressure (left) and flux (right)
• Figure 5: Number of iterations until convergence for slice 42: pressure (left) and flux (right).