In this talk we will present a complete set of fully discretized integral operators for the Helmholtz equation in two dimensions, working on any fine set of smooth closed boundaries. The goal of this collection is the development of do-it-yourself tools for simultaneous use of all boundary integral operators, with automatic handling of complicated geometries, while based of very simple-minded discretization techniques that lead to extremely easy coding (with a level of difficulty not much higher than one dimensional finite differences).
The discretization is carried out by carefully choosing source and observation points, leading to a type of Nystrom approximation of all operators. The choices depend on several parameters: staggering of the grids, symmetrization-and-stabilization coefficients, and quadrature weights.
I will comment on how the existing asymptotic results that justified the tilted Calderon Calculus of order two can be combined to create a fully discrete set of integral equations that are of order three in all quantities, meaning that potentials are recovered with order three, but also boundary quantities are obtained with uniform order three approximations.
I will finally show how developing the same calculus for the Laplace resolvent equation leads to a fully discrete method for the transient wave equation, by combining the Nystrom matrices with convolution quadrature.