Efficient numerical method for reliable upper and lower bounds on homogenized parameters

TL;DR

This paper introduces a numerical method that guarantees reliable upper and lower bounds on the effective parameters of elliptic PDEs, especially in 3D, using finite element approximations for both primal and dual problems.

Contribution

It presents a novel finite element-based approach to compute guaranteed bounds on homogenized coefficients in 3D elliptic problems, including theoretical justification and practical examples.

Findings

Provides guaranteed two-sided bounds on effective coefficients

Extends the methodology to 3D elliptic problems

Demonstrates effectiveness with illustrative examples

Abstract

A numerical procedure providing guaranteed two-sided bounds on the effective coefficients of elliptic partial differential operators is presented. The upper bounds are obtained in a standard manner through the variational formulation of the problem and by applying the finite element method. To obtain the lower bounds we formulate the dual variational problem and introduce appropriate approximation spaces employing the finite element method as well. We deal with the 3D setting, which has been rarely considered in the literature so far. The theoretical justification of the procedure is presented and supported with illustrative examples.