  We present a non-nested multilevel algorithm for solving the Poisson equation
discretized at scattered points using polyharmonic radial basis function
(PHS-RBF) interpolations. We append polynomials to the radial basis functions
to achieve exponential convergence of discretization errors. The interpolations
are performed over local clouds of points and the Poisson equation is
collocated at each of the scattered points, resulting in a sparse set of
discrete equations for the unkown variables. To solve this set of equations, we
have developed a non-nested multilevel algorithm utilizing multiple
independently generated coarse sets of points. The restriction and prolongation
operators are also constructed with the same RBF interpolations procedure. The
performance of the algorithm for Dirichlet and all-Neumann boundary conditions
is evaluated in three model geometries using a manufactured solution. For
Dirichlet boundary conditions, rapid convergence is observed using SOR point
solver 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, the
convergence is seen to significantly improve. The GMRES accelerated multilevel
algorithm is included in a fractional step method to solve incompressible
Navier-Stokes equations.
