# On jump relations of anisotropic elliptic interface problems

## TL;DR

This paper develops a systematic approach to derive interface relations for anisotropic elliptic PDEs with discontinuities, addressing the challenges posed by anisotropy and coordinate transformations, which is crucial for high-order numerical methods.

## Contribution

It introduces a new systematic method for deriving interface relations in anisotropic elliptic problems, extending beyond isotropic cases and handling coordinate transformation issues.

## Key findings

- Derived interface relations for anisotropic elliptic PDEs in 2D and 3D.
- Addressed invariance issues under coordinate transformations.
- Facilitated development of high-order numerical methods.

## Abstract

Almost all materials are anisotropic. In this paper, interface relations of anisotropic elliptic partial differential equations involving discontinuities across interfaces are derived in two and three dimensions. Compared with isotropic cases, the invariance of partial differential equations and the jump conditions under orthogonal coordinates transformation is not valid anymore. A systematic approach to derive the interface relations is established in this paper for anisotropic elliptic interface problems, which can be important for deriving high order accurate numerical methods.