Solving the Dirichlet problem for the Poisson equation by the least squares collocation method with given discrete boundary domain
The paper addresses a new version of the least squares collocation (LSC) method proposed and implemented for the numerical solution of boundary value problems for the Poisson's equation in case of given discrete domain boundary. The computer program builds a continuous double spline when boundary is smooth or a piecewise smooth double spline if boundary has salient points. In this version of the method we apply the idea of using parts of the cells of a regular grid (outside the domain). These parts of the cells are cut off by the boundary to construct the LSC method. It is assumed that the solution has no singularities at the boundary and in a certain small neighborhood of it. The differential equation for the problem is correct not only in the computational domain but also in a small neighborhood of its boundary. Then the idea of attaching “small” irregular cells to neighboring ones is used in the work with the aim of reducing the number of conditionality of the global system of linear algebraic equations. It is shown that the approximate solutions obtained by the LSC method converge with an increased order and coincide with the analytical solutions of the test problems with high accuracy.