Domain decomposition finite element/finite difference method for the conductivity reconstruction in a hyperbolic equation

TL;DR

This paper introduces a hybrid domain decomposition method combining finite element and finite difference techniques to efficiently solve hyperbolic equations, demonstrated on 3D conductivity reconstruction problems.

Contribution

The paper presents a novel hybrid finite element/finite difference domain decomposition approach with explicit schemes that ensure stability at interfaces for hyperbolic equations.

Findings

Efficient conductivity reconstruction in 3D hyperbolic equations.

Stable interface handling between finite element and finite difference regions.

Improved computational efficiency through domain decomposition.

Abstract

We present domain decomposition finite element/finite difference method for the solution of hyperbolic equation. The domain decomposition is performed such that finite elements and finite differences are used in different subdomains of the computational domain: finite difference method is used on the structured part of the computational domain and finite elements on the unstructured part of the domain. The main goal of this method is to combine flexibility of finite element method and efficiency of a finite difference method. An explicit discretization schemes for both methods are constructed such that finite element and finite difference schemes coincide on the common structured overlapping layer between computational subdomains. Then the resulting scheme can be considered as a pure finite element scheme which allows avoid instabilities at the interfaces. We illustrate efficiency of the domain decomposition method on the reconstruction of the conductivity function in the hyperbolic equation in three dimensions.