# Solving Elliptic Interface Problems with Jump Conditions on Cartesian Grids

## TL;DR

This paper introduces a straightforward numerical method for elliptic equations with discontinuities across irregular interfaces, achieving high accuracy and stable condition numbers on Cartesian grids in 2D and 3D.

## Contribution

The paper presents a simple, second-order accurate algorithm for elliptic interface problems with jump conditions on Cartesian grids, maintaining bounded condition numbers regardless of diffusion coefficient ratios.

## Key findings

- Achieves second-order accuracy in solutions
- Attains first-order accuracy in gradients in the $L^
abla$-norm
- Condition number remains bounded regardless of diffusion coefficient ratio

## Abstract

We present a simple numerical algorithm for solving elliptic equations where the diffusion coefficient, the source term, the solution and its flux are discontinuous across an irregular interface. The algorithm produces second-order accurate solutions and first-order accurate gradients in the $L^\infty$-norm on Cartesian grids. The condition number is bounded, regardless of the ratio of the diffusion constant and scales like that of the standard 5-point stencil approximation on a rectangular grid with no interface. Numerical examples are given in two and three spatial dimensions.