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We present a non-nested multilevel algorithm for solving the Poisson equationdiscretized at scattered points using polyharmonic radial basis function(PHS-RBF) interpolations. We append polynomials to the radial basis functionsto achieve exponential convergence of discretization errors. The interpolationsare performed over local clouds of points and the Poisson equation iscollocated at each of the scattered points, resulting in a sparse set ofdiscrete equations for the unkown variables. To solve this set of equations, wehave developed a non-nested multilevel algorithm utilizing multipleindependently generated coarse sets of points. The restriction and prolongationoperators are also constructed with the same RBF interpolations procedure. Theperformance of the algorithm for Dirichlet and all-Neumann boundary conditionsis evaluated in three model geometries using a manufactured solution. ForDirichlet boundary conditions, rapid convergence is observed using SOR pointsolver as the relaxation scheme. For cases of all-Neumann boundary conditions,convergence is seen to slow down with the degree of the appended polynomial.However, when the multilevel procedure is combined with a GMRES algorithm, theconvergence is seen to significantly improve. The GMRES accelerated multilevelalgorithm is included in a fractional step method to solve incompressibleNavier-Stokes equations.

A Non-Nested Multilevel Method for Meshless Solution of the Poisson Equation in Heat Transfer and Fluid Flow