Iterative oversampling technique for constraint energy minimizing generalized multiscale finite element method in the mixed formulation

Highlights

• A new iterative oversampling technique for Darcy flow in heterogeneous and high contrast media.

• The convergence of the proposed method is exponential with respect to the number of iterations.

• A rigorous proof is provided for the proposed method.

Abstract

In this paper, we develop an iterative scheme to construct multiscale basis functions within the framework of the Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM) for the mixed formulation. The iterative procedure starts with the construction of an energy minimizing snapshot space that can be used for approximating the solution of the model problem. A spectral decomposition is then performed on the snapshot space to form global multiscale space. Under this setting, each global multiscale basis function can be split into a non-decaying and a decaying parts. The non-decaying part of a global basis is localized and it is fixed during the iteration. Then, one can approximate the decaying part via a modified Richardson scheme with an appropriately defined preconditioner. Using this set of iterative-based multiscale basis functions, first-order convergence with respect to the coarse mesh size can be shown if sufficiently many times of iterations with regularization parameter being in an appropriate range are performed. Numerical results are presented to illustrate the effectiveness and efficiency of the proposed computational multiscale method.

Introduction

Many problems arising from engineering involve heterogeneous materials which have strong contrasts in their physical properties. In general, one may model these so-called multiscale problems using partial differential equations (PDEs) with high-contrast valued multiscale coefficients. An important example is Darcy’s law describing flow in highly heterogeneous porous media. These problems are prohibitively costly to solve when traditional fine-scale solvers are directly applied. The direct simulation of multiscale PDEs with accurate resolution can be costly as a relatively fine mesh is required to resolve the coefficients, leading to a prohibitively large number of degrees of freedom, a high percentage of which may be extraneous. Therefore, some types of model-order reductions are necessary to avoid high computational cost in simulation.

These computational challenges have been addressed by the development of efficient model reduction techniques and many model reduction techniques have been well explored in existing literature. For instance, in upscaling methods [7], [20], [26], [46] which are commonly used, one typically derives upscaling media based on the model problem and solves the resulting upscaled problem globally on a coarser grid. In general, this derivation can be done by solving a class of local cell problems in the coarse elements. Besides upscaling approaches mentioned above, multiscale methods [22], [23], [28], [29] have been widely used to approximate the solution of the multiscale problem. In multiscale methods, the solution to the problem is approximated using local basis functions, which are solutions to a class of local problems, which are related to the model, on the coarse grid. Moreover, because of the necessity of the mass conservation for velocity fields, many approaches have been proposed to guarantee this property, such as multiscale finite volume methods [19], [27], [30], [34], [35], [37], mixed multiscale finite element methods (MsFEM) [1], [2], [8] and its generalization GMsFEM [5], [6], [12], [13], [15], [18], mortar multiscale methods [3], [41], [42], [47] and various post-processing methods [4], [38].

Among these multiscale methods mentioned above, we focus on the framework of GMsFEM in this work. The GMsFEM in mixed formulation has been developed in [13], [24] recently and it provides a systematic procedure to construct multiple basis functions for either velocity or pressure in each local patch, which makes these methods different to previous methodology in applications. The computation of velocity basis functions involves a construction of snapshot space and a model reduction via local spectral decomposition to identify appropriate modes to form the multiscale space. The convergence analysis in [13] addresses a spectral convergence with convergence rate proportional to ${\Lambda }^{-1}$, where $\Lambda$ is the smallest eigenvalue whose modes are excluded in the multiscale space. In [14], a variation of GMsFEM based on a constraint energy minimization (CEM) strategy for mixed formulation has been developed. This approach is inspired by the work on localization [36], [39], [40] and makes use of the ideas of oversampling to compute multiscale basis functions in oversampled subregions with the satisfaction of an appropriate orthogonality condition, where similar ideas have been applied for various numerical discretization and model problems [9], [10], [11], [16], [17], [33]. The method proposed in [14] provides a mass conservative velocity field and allows one to identify some non-local information depending on the inputs of the problem. One can show that the CEM-GMsFEM provides a better convergence rate (comparing to the mixed GMsFEM) that is proportional to $H{\Lambda }^{-1}$ with $H$ the size of coarse mesh if the size of oversampling regions is at least of the logarithmic magnitude of the product of the coarse mesh size and the value of contrast.

However, in the original framework of CEM-GMsFEM, one needs to construct the multiscale basis functions supported in relatively large oversampling regions in order to guarantee a certain level of accuracy and it leads to a moderately large computational cost in the offline stage. In particular, when dealing with the case of high-contrast permeability, one needs to set the oversampling parameters to be the logarithm of the value of contrast and it results in a loss of sparsity of the stiffness and mass matrices.

In this work, we propose an iterative computational scheme to construct multiscale basis functions (for velocity) satisfying the property of CEM to overcome the issue mentioned above and enhance the computational efficiency. The proposed method relates to the theory of iterative solvers and subspace decomposition methods [31], [32], [43] (see also the discussion in [25, Remark 2.6] about the iterative implementation of a class of numerical homogenization methods). For the mixed formulation, the construction of multiscale basis functions in this work slightly differs from that of the original CEM-GMsFEM and it starts with a set of localized energy minimizing snapshot functions. Conceptually, we decompose the global multiscale basis function with CEM property into a decaying and a non-decaying parts. The non-decaying part is formed by the energy minimizing snapshots thus it is localized and it will be fixed during the iterations. Then, starting with a zero initial condition, we approximate the decaying part using an iterative scheme of the type of modified Richardson [44], [45] (or any other iterative methods) with an appropriate designed preconditioner. The size of support of the approximated decaying part is proportional to the number of iterations, which can be freely adjusted by the user. Hence, this (iterative) construction for multiscale basis functions is more flexible than the one in the original CEM-GMsFEM. The iterative process will maintain the property of mass conservation and no need for any post-processing techniques. The proposed method has an advantage that the marginal computational cost from one iteration to the next one is comparatively low. This iterative construction also shows some potential to compute the offline CEM basis functions in an adaptive manner to further reduce the cost of computation. With careful selection of regularization parameter in the iterative scheme, one can show the first-order convergence rate (with respect to $H$) of the velocity if sufficiently many iteration times, depending only on the coarse mesh and the quantity $\Lambda$ in GMsFEM, are performed in the offline stage.

The paper is organized as follows. In Section 2, we present some preliminaries of the model problem considered in this work. We also briefly review the framework of the original CEM-GMsFEM. Then, we derive the iterative construction of multiscale basis functions for velocity in Section 3. Next, we provide a complete analysis of the proposed iterative construction in Section 4. In particular, we estimate the condition number of the matrix in the iteration in Lemma 4.6. The main theoretical results of the sufficient condition of linear convergence reads in Theorem 4.1. Several numerical tests are provided in Section 5 to demonstrate the performance of the numerical methods based on the iterative scheme. Finally, some concluding remarks are drawn in Section 6.