Non-intrusive implementation of a wide variety of Multiscale Finite Element Methods

TL;DR

This paper presents a non-intrusive framework for implementing various Multiscale Finite Element Methods, making them more accessible for industrial applications and enhancing their theoretical understanding.

Contribution

It introduces an abstract, general framework enabling non-intrusive implementation of diverse MsFEMs for linear PDEs, facilitating software integration and analysis.

Findings

Developed a unified non-intrusive implementation approach

Enhanced dissemination of MsFEMs in industrial settings

Provided insights into theoretical analysis of MsFEMs

Abstract

Multiscale Finite Element Methods (MsFEMs) are now well-established finite element type approaches dedicated to multiscale problems. They first compute local, oscillatory, problem-dependent basis functions that generate a suitable discretization space, and next perform a Galerkin approximation of the problem on that space. We investigate here how these approaches can be implemented in a non-intrusive way, in order to facilitate their dissemination within industrial codes or non-academic environments. We develop an abstract framework that covers a wide variety of MsFEMs for linear second-order partial differential equations. Non-intrusive MsFEM approaches are developed within the full generality of this framework, which may moreover be beneficial to steering software development and improving the theoretical understanding and analysis of MsFEMs.