Reduced Basis Multiscale Finite Element Methods for Elliptic Problems

Abstract

Abstract.In this paper, we propose reduced basis multiscale finite element methods (RB-MsFEM) for elliptic problems with highly oscillating coefficients. The method is based on multiscale finite element methods with local test functions that encode the oscillatory behavior ([4, 14]). For uniform rectangular meshes, the local oscillating test functions are represented by a reduced basis method, parameterizing the center of the elements. For triangular elements, we introduce a slightly different approach. By exploring over-sampling of the oscillating test functions, initially introduced to recover a better approximations of the global harmonic coordinate map, we first build the reduced basis on uniform rectangular elements containing the original triangular elements and then restrict the oscillating test function to the triangular elements. These techniques are also generalized to the case where the coefficients dependent on additional independent parameters. The analysis of the proposed methods is supported by various numerical results, obtained on regular and unstructured grids.Key words. Multiscale finite element methods, reduced basis methods.AMS subject classifications. TBD 1. Introduction. The development of efficient and accurate numerical methods for solving problems with highly oscillating coefficients is an area of research that is increasingly active. This is not only driven by applications in subsurface flows or the modeling of novel materials, but also by the prohibitive cost of solving such problems using a straightforward approach where all scales are adequately resolved.To address and, ultimately overcome, the computational cost of resolving the finest scale, multiscale finite element methods (MsFEM) have been developed in [15,16,10,17,9]. In this approach, accuracy is achieved by locally solving a fine scale problem. These solutions are subsequently used to build the multiscale finite element basis, encoding the local fine structure, to capture the fine scale information of the leading order differential operator. Originally, MsFEM was proposed for linear finite elements but generalized in [4] to enable the use of high-order elements by local oscillating test functions (or harmonic coordinates). In [14], we proposed an alternative formulation of high-order MsFEM using related ideas, albeit introducing a more natural formulation.The local oscillating test functions must be solved for each element of the coarse mesh and each of these local problems must be resolve to fully capture the local fine scale. Hence, these local problems has many degrees of freedom (DOFs) and a fast solver of these local problems is central to the efficiency of MsFEM. To address this, we consider the reduced basis method (RBM) as an ideal technique that provides an efficient representation of the solution to parametric problems in manyquery and real-time scenarios, see introductions in [23,24, 25]. This idea has been pursued by a few authors recently in the context of multi-scale problems, as in [7], where reduced basis met...

Randomized Local Model Order Reduction

“…In order to address parameterized multiscale problems the local approximation spaces are for instance spanned by eigenfunctions of an eigenvalue problem on the space of harmonic functions in [28], generated by solving the global parameterized PDE and restricting the solution to the respective subdomain in [70,3], or enriched in the online stage by local solutions of the PDE, prescribing the insufficient RB solution as Dirichlet boundary conditions in [70,3]. Apart from that the RB method has also been used in the context of multiscale methods for example in [69,40,1].…”

The localized reduced basis multiscale method for two‐phase flows in porous media

“…Furthermore, in [48,49], the RB was applied in the context of heterogeneous multiscale methods to reduce the overall workload by replacing the costly, high-dimensional micro problems by reduced surrogates parametrized by the center of the macro quadrature points. In [50], the same idea was applied in the context of multiscale FEMs, and in [51], the reduced basis method is used to efficiently compute solutions to fine-scale equilibrium problems in representative volume elements in the context of multiscale homogenization.…”

Displacement-based multiscale modeling of fiber-reinforced composites by means of proper orthogonal decomposition

“…Therefore, the unknown displacements of the RVE boundary value problem (14) are approximated by (22) where is the n f × m-dimensional subspace matrix. In this way the n f -dimensional displacement vector U f is reduced to the m-dimensional unknown vector U red .…”

“…By means of the derivation of upper and lower error bounds, the authors enable adaptive computations of random linear-elastic composites. The coupling of a reduced basis method to multiscale finite element methods for elliptic problems with highly oscillating coefficients can be found in Hesthaven et al [22]. A finite element-based heterogeneous multiscale method applied to crack domains has been published in Abdulle and Bai [23].…”