Localized Harmonic Characteristic Basis Functions for Multiscale Finite Element Methods

TL;DR

This paper introduces a multiscale finite element method using localized harmonic characteristic basis functions to efficiently solve elliptic equations in highly heterogeneous materials with discontinuous coefficients.

Contribution

It develops a novel multiscale finite element approach based on asymptotic expansions tailored for heterogeneous materials with inclusions.

Findings

Method achieves accurate approximations in complex heterogeneous media

Numerical results demonstrate computational efficiency and robustness

Basis functions effectively capture local heterogeneities

Abstract

We solve elliptic systems of equations posed on highly heterogeneous materials. Examples of this class of problems are composite structures and geological processes. We focus on a model problem which is a second-order elliptic equation with discontinuous coefficients. These coefficients represent the conductivity of a composite material. We assume a background with low conductivity that contains inclusions with different thermal properties. Under this scenario we design a multiscale finite element method to efficiently approximate solutions. The method is based on an asymptotic expansions of the solution in terms of the ratio between the conductivities. The resulting method constructs (locally) finite element basis functions (one for each inclusion). These bases that generate the multiscale finite element space where the approximation of the solution is computed. Numerical experiments show the good performance of the proposed methodology.