Mixed Generalized Multiscale Finite Element Method for Flow Problem in Thin Domains

TL;DR

This paper develops a Mixed Generalized Multiscale Finite Element Method for efficiently approximating flow problems in thin 2D domains, using multiscale basis functions and spectral problems to reduce computational complexity.

Contribution

It introduces a novel multiscale method with basis functions based on local snapshot spaces and spectral problems, tailored for flow in thin domains, with convergence analysis and numerical validation.

Findings

The method achieves accurate approximations with fewer basis functions.

Numerical results confirm convergence and effectiveness in complex geometries.

The approach handles both homogeneous and heterogeneous properties effectively.

Abstract

In this paper, we construct a class of Mixed Generalized Multiscale Finite Element Methods for the approximation on a coarse grid for an elliptic problem in thin two-dimensional domains. We consider the elliptic equation with homogeneous boundary conditions on the domain walls. For reference solution of the problem, we use a Mixed Finite Element Method on a fine grid that resolves complex geometry on the grid level. To construct a lower dimensional model, we use the Mixed Generalized Multiscale Finite Element Method, which is based on some multiscale basis functions for velocity fields. The construction of the basis functions is based on the local snapshot space that takes all possible flows on the interface between coarse cells into account. In order to reduce the size of the snapshot space and obtain the multiscale approximation, we solve a local spectral problem to identify dominant modes in the snapshot space. We present a convergence analysis of the presented multiscale method. Numerical results are presented for two-dimensional problems in three testing geometries along with the errors associated to different numbers of the multiscale basis functions used for the velocity field. Numerical investigations are conducted for problems with homogeneous and heterogeneous properties respectively.