Multiscale empirical interpolation for solving nonlinear PDEs

JOURNAL OF COMPUTATIONAL PHYSICS

卷:278

Article

Calo, Victor M.^1,2  Efendiev, Yalchin^1,3  Galvisd, Juan⁴  Ghommem, Mehdi¹ 

[1] King Abdullah Univ Sci & Technol, Ctr Numer Porous Media NumPor, Thuwal 239556900, Saudi Arabia

[2] King Abdullah Univ Sci & Technol, Thuwal 239556900, Saudi Arabia

[3] Texas A&M Univ, ISC, College Stn, TX 77843 USA

[4] Univ Nacl Colombia, Dept Matemat, Bogota, Colombia

关键词: Generalized multiscale finite element methods;    Discrete empirical interpolation method;    Model reduction;    Nonlinear PDEs;   

DOI  :  10.1016/j.jcp.2014.07.052

来源: Elsevier

【 摘 要 】

In this paper, we propose a multiscale empirical interpolation method for solving nonlinear multiscale partial differential equations. The proposed method combines empirical interpolation techniques and local multiscale methods, such as the Generalized Multiscale Finite Element Method (GMsFEM). To solve nonlinear equations, the GMsFEM is used to represent the solution on a coarse grid with multiscale basis functions computed offline. Computing the GMsFEM solution involves calculating the system residuals and Jacobians on the fine grid. We use empirical interpolation concepts to evaluate these residuals and Jacobians of the multiscale system with a computational cost which is proportional to the size of the coarse-scale problem rather than the fully-resolved fine scale one. The empirical interpolation method uses basis functions which are built by sampling the nonlinear function we want to approximate a limited number of times. The coefficients needed for this approximation are computed in the offline stage by inverting an inexpensive linear system. The proposed multiscale empirical interpolation techniques: (1) divide computing the nonlinear function into coarse regions; (2) evaluate contributions of nonlinear functions in each coarse region taking advantage of a reduced-order representation of the solution; and (3) introduce multiscale proper-orthogonal-decomposition techniques to find appropriate interpolation vectors. We demonstrate the effectiveness of the proposed methods on several nonlinear multiscale PDEs that are solved with Newton's methods and fully-implicit time marching schemes. Our numerical results show that the proposed methods provide a robust framework for solving nonlinear multiscale PDEs on a coarse grid with bounded error and significant computational cost reduction. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free