Mesoscale asymptotic approximations to solutions of mixed boundary value problems in perforated domains

TL;DR

This paper develops asymptotic approximation methods for solving mixed boundary value problems involving the Laplacian in three-dimensional perforated domains with many small voids, providing explicit formulas and solvability conditions.

Contribution

It introduces a novel asymptotic approximation technique using dipole fields for Laplacian problems in perforated domains with arbitrary-shaped voids, including solvability analysis.

Findings

The linear algebraic system for coefficients is solvable.

An energy estimate for the approximation remainder is derived.

The method applies to voids with diameters smaller than their separation distance.

Abstract

We describe a method of asymptotic approximations to solutions of mixed boundary value problems for the Laplacian in a three-dimensional domain with many perforations of arbitrary shape, with the Neumann boundary conditions being prescribed on the surfaces of small voids. The only assumption made on the geometry is that the diameter of a void is assumed to be smaller compared to the distance to the nearest neighbour. The asymptotic approximation, obtained here, involves a linear combination of dipole fields constructed for individual voids, with the coefficients, which are determined by solving a linear algebraic system. We prove the solvability of this system and derive an estimate for its solution. The energy estimate is obtained for the remainder term of the asymptotic approximation.