[Paper Review] Generalized multiscale finite element methods for space-time heterogeneous parabolic equations

Eric T. Chung, Yalchin Efendiev+2 more|arXiv (Cornell University)|May 24, 2016

Advanced Mathematical Modeling in Engineering15 references3 citations

TL;DR

This paper introduces a generalized multiscale finite element method (GMsFEM) for space-time heterogeneous parabolic equations using space-time coarse cells. It constructs space-time snapshot and offline spaces via local problems with randomized boundary conditions and oversampling, enabling accurate solutions with low-dimensional coarse spaces, and demonstrates fast convergence using online basis functions enriched with residual information.

ABSTRACT

In this paper, we consider local multiscale model reduction for problems with multiple scales in space and time. We developed our approaches within the framework of the Generalized Multiscale Finite Element Method (GMsFEM) using space-time coarse cells. The main idea of GMsFEM is to construct a local snapshot space and a local spectral decomposition in the snapshot space. Previous research in developing multiscale spaces within GMsFEM focused on constructing multiscale spaces and relevant ingredients in space only. In this paper, our main objective is to develop a multiscale model reduction framework within GMsFEM that uses space-time coarse cells. We construct space-time snapshot and offline spaces. We compute these snapshot solutions by solving local problems. A complete snapshot space will use all possible boundary conditions; however, this can be very expensive. We propose using randomized boundary conditions and oversampling. We construct the local spectral decomposition based on our analysis, as presented in the paper. We present numerical results to confirm our theoretical findings and to show that using our proposed approaches, we can obtain an accurate solution with low dimensional coarse spaces. We remark that the proposed method is a significant extension compared to existing methods, which use coarse cells in space only because of (1) the parabolic nature of cell solutions, (2) extra degrees of freedom associated with space-time cells, and (3) local boundary conditions in space-time cells.

Motivation & Objective

• To develop a systematic multiscale model reduction framework for space-time heterogeneous parabolic problems with non-separable scales.
• To extend GMsFEM beyond spatial-only coarse cells by incorporating time into the multiscale basis construction.
• To address challenges from the parabolic nature of solutions, increased degrees of freedom in space-time cells, and complex local boundary conditions.
• To enable accurate and efficient solution of high-contrast, multiscale parabolic problems using low-dimensional coarse spaces.

Proposed method

• Constructs space-time snapshot spaces by solving local problems over space-time coarse blocks with all possible boundary conditions.
• Applies randomized boundary conditions and oversampling to reduce computational cost of full snapshot space construction.
• Performs local spectral decomposition on the snapshot space to extract dominant modes for offline space construction.
• Uses online basis functions enriched with residual information to correct the solution and accelerate convergence.
• Employs a space-time variational formulation with time-dependent coefficients to capture coupled space-time dynamics.
• Integrates offline and online stages: offline for basis generation, online for adaptive enrichment using residual-driven corrections.

Experimental results

Research questions

• RQ1 Can a systematic space-time multiscale model reduction framework be developed within the GMsFED framework for parabolic equations with multiple scales in space and time?
• RQ2 How can snapshot and offline spaces be constructed effectively in space-time coarse cells to capture multiscale features?
• RQ3 What is the impact of randomized boundary conditions and oversampling on the accuracy and efficiency of the space-time GMsFEM?
• RQ4 How do online basis functions based on residual information improve convergence in the online stage?
• RQ5 What is the theoretical and numerical performance of the proposed method for high-contrast, heterogeneous parabolic problems?

Key findings

• The proposed space-time GMsFEM achieves accurate solutions using low-dimensional coarse spaces, significantly reducing computational cost compared to fine-scale simulations.
• Randomized boundary conditions and oversampling enable efficient construction of snapshot spaces without requiring all possible boundary conditions.
• Online basis functions, derived from residual information, accelerate convergence and improve accuracy when combined with sufficient offline basis functions.
• Numerical results confirm the theoretical convergence rates and demonstrate robust performance for problems with high-contrast and non-separable scales.
• The method outperforms existing approaches that use only spatial coarse cells, especially in problems with strong temporal heterogeneity and coupled space-time dynamics.