A Collocation Method with Exact Imposition of Mixed Boundary Conditions

In this paper, we propose a natural collocation method with exact imposition of mixed boundary conditions based on a generalized Gauss-Lobatto-Legendre-Birhoff quadrature rule that builds in the underlying boundary data. We provide a direct construction of the quadrature rule, and show that the collocation method can be implemented as efficiently as the usual collocation scheme for PDEs with Dirichlet boundary conditions. We apply the collocation method to some model PDEs and the time-harmonic Helmholtz equation, and demonstrate its spectral accuracy and efficiency by various numerical examples.