Meshless Poisson Problems in the Finite Pointset Method: Positive Stencils and Multigrid

The finite pointset method is a meshless Lagrangian particle method. In the application to incompressible viscous fluid flow the solution of Poisson problems on the cloud of particles is a fundamental subproblem. A valuable property of finite difference approximations to the Laplace operator is the positivity of stencils, i.e., all weights of neighboring points are positive. Classical least squares approaches do not guarantee positive stencils. We present a new approach, based on linear minimization, which enforces positivity of stencils and additionally yields a minimal number of nonzero stencil entries. The resulting system matrices are M-matrices, which is of particular interest with respect to multigrid solvers.