Regularized integral formulation of mixed Dirichlet-Neumann problems

TL;DR

This paper introduces a new integral equation approach for solving mixed Dirichlet-Neumann boundary problems on smooth 2D domains, providing high-order accurate solutions by addressing singularities with novel regularization and quadrature techniques.

Contribution

It offers the first detailed theoretical analysis of Zaremba boundary singularities and develops high-order numerical algorithms using Green functions, integral equations, and Fourier Continuation methods.

Findings

Algorithms achieve high-order convergence.

Effective handling of Zaremba boundary singularities.

Applicable to Helmholtz and Laplace problems with high accuracy.

Abstract

This paper presents a theoretical discussion as well as novel solution algorithms for problems of scattering on smooth two-dimensional domains under Zaremba boundary conditions for which Dirichlet and Neumann conditions are specified on various portions of the domain boundary. The theoretical basis of the proposed numerical methods, which is provided for the first time in the present contribution, concerns detailed information about the singularity structure of solutions of the Helmholtz operator under boundary conditions of Zaremba type. The new numerical method is based on use of Green functions and integral equations, and it relies on the Fourier Continuation method for regularization of all smooth-domain Zaremba singularities as well as newly derived quadrature rules which give rise to high-order convergence even around Zaremba singular points. As demonstrated in this paper, the resulting algorithms enjoy high-order convergence, and they can be used to efficiently solve challenging Helmholtz boundary value problems and Laplace eigenvalue problems with high-order accuracy.