# An integral equation based numerical method for the forced heat equation on complex domains

## TL;DR

This paper develops an integral equation based numerical method for the heat equation on complex domains, combining time discretization, boundary integral solutions, and volume potentials for inhomogeneous sources.

## Contribution

It introduces a novel approach that extends integral equation methods to inhomogeneous and time-dependent heat equations, including efficient function extension and specialized quadrature techniques.

## Key findings

- Achieves high accuracy in solving the heat equation on complex geometries.
- Effectively handles inhomogeneous source terms with volume potentials.
- Provides a stable and efficient numerical framework for time-dependent PDEs.

## Abstract

Integral equation based numerical methods are directly applicable to homogeneous elliptic PDEs, and offer the ability to solve these with high accuracy and speed on complex domains. In this paper, extensions to problems with inhomogeneous source terms and time dependent PDEs, such as the heat equation, have been introduced. One such approach for the heat equation is to first discretise in time, and in each time-step solve a so-called modified Helmholtz equation with a parameter depending on the time step size. The modified Helmholtz equation is then split into two parts: a homogeneous part solved with a boundary integral method and a particular part, where the solution is obtained by evaluating a volume potential over the inhomogeneous source term over a simple domain. In this work, we introduce two components which are critical for the success of this approach: a method to efficiently compute a high-regularity extension of a function outside the domain where it is defined, and a special quadrature method to accurately evaluate singular and nearly singular integrals in the integral formulation of the modified Helmholtz equation for all time step sizes.