# Software frameworks for integral equations in electromagnetic scattering based on Calder\'on identities

## TL;DR

This paper presents a software framework based on Calderón identities within the BEM++ library that simplifies the formulation and implementation of stable boundary integral equations for electromagnetic scattering problems.

## Contribution

It introduces a robust, efficient implementation of Calderón identities and a product algebra that automates the assembly of Maxwell boundary integral equations in software.

## Key findings

- Framework enables simple formulation of Maxwell boundary integral equations.
- Implementation improves robustness and efficiency of boundary element methods.
- Demonstrates practical use in deriving standard electromagnetic integral equations.

## Abstract

In recent years there have been tremendous advances in the theoretical understanding of boundary integral equations for Maxwell problems. In particular, stable dual pairing of discretisation spaces have been developed that allow robust formulations of the preconditioned electric field, magnetic field and combined field integral equations. Within the BEM++ boundary element library we have developed implementations of these frameworks that allow an intuitive formulation of the typical Maxwell boundary integral formulations within a few lines of code. The basis of these developments is an efficient and robust implementations of Calder\'on identities together with a product algebra that hides and automates most technicalities involved in assembling Galerkin boundary integral equations. In this paper we demonstrate this framework and use it to derive very simple and robust software formulations of the standard preconditioned electric field, magnetic field and regularised combined field integral equations for Maxwell.