A meshless point collocation solver for elliptic boundary value problems

We propose a strong-form meshless point collocation (MPC) solver for the Poisson equation. The Poisson equation allows us to compute approximate pressure corrections and ensure the incompressibility of the velocity field or to solve for the stream function and vorticity. We discretize the spatial domain using quadratic triangular elements in 2D and tetrahedral elements in 3D. The element nodes, including the vertices and edge midpoints, define the point cloud used in the MPC method. We determine the support domain for each using the mesh’s connectivity. When constructing the stiffness matrix, the resulting algebraic systems have the same bandwidth as those generated by the finite element (FE) method. We use direct and iterative solvers to assess the accuracy and efficiency of the MPC method. Our solution strategy enables automatic mesh generation, as nodes are utilized directly in the interpolation construction, eliminating the need to evaluate mesh quality. Finally, we investigate the efficiency of the MPC method in solving linear systems in 3D with a large number of nodes.