Solution of parameter-dependent diffusion equation in layered media

TL;DR

This paper derives explicit approximate solutions for parameter-dependent diffusion equations in layered media, enabling efficient computation of solution maps with applications to reduced basis methods.

Contribution

It introduces explicit solution formulas for layered diffusion problems, extending from two layers to multiple layers, and analyzes their application in reduced basis methods.

Findings

Explicit three-term solution formula for two-layer case.

Extension of the formula to multi-layer media.

Numerical examples demonstrating the approach's effectiveness.

Abstract

This work studies the parameter-dependent diffusion equation in a two-dimensional domain consisting of locally mirror symmetric layers. It is assumed that the diffusion coefficient is a constant in each layer. The goal is to find approximate parameter-to-solution maps that have a small number of terms. It is shown that in the case of two layers one can find a solution formula consisting of three terms with explicit dependencies on the diffusion coefficient. The formula is based on decomposing the solution into orthogonal parts related to both of the layers and the interface between them. This formula is then expanded to an approximate one for the multi-layer case. We give an analytical formula for square layers and use the finite element formulation for more general layers. The results are illustrated with numerical examples and have applications for reduced basis methods by analyzing the Kolmogorov n-width.