  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.
