# Open boundary Dirichlet problems for Laplace's equation in the plane

In standard numerical solvers for the Navier-Stokes equations, a Poisson problem must be solved on a uniform grid at each time step. Typically, the pressure or some related quantity is determined from this problem, and, in order to achieve high accuracy, careful treatment must be taking when solving the problem. This is especially true if an elastic structure is submerged in the flow because the pressure may be discontinous or have discontinous gradients at the structure. In applications in which such a structure is closed, methods have been developed that lead to the correct behavior of the solution. Jump conditions, or other information about the behavior at the structure is used to modify a standard method in order to capture the correct behavior. However, in applications with open immersed structures, such as swimming organisms, the literature does not contain much information about proper behavior near the structure This dissertation works towards filling these gaps in the literature. As a model of the situation, functions which are harmonic in the planar compliment of an arc, and with Dirichlet conditions on the arc are considered. For the cases of line segments and arcs of circles, specific properties of the behavior of such functions are determined using recent work by Jiang and Rokhlin The numerical method presented is a modified finite difference scheme with additional treatment at grid nodes near the boundary. Such treatment is necessary since the gradients of the solution are discontinuous across the boundary and unbounded at the endpoints Several examples are given. The examples show that applying a standard finite difference method does not lead to proper convergence. This fact is linked to the singular endpoint behavior of the solution. However, other examples show that the method developed in this dissertation has the same order of accuracy as the finite difference method has when applied to approximate smooth functions