We investigate the influence of the shape parameter in the meshless Gaussian radial basis
function finite difference (RBF-FD) method with irregular centres on the quality of the
approximation of the Dirichlet problem for the Poisson equation with smooth solution.
Numerical experiments show that the optimal shape parameter strongly depends on
the problem, but insignificantly on the density of the centres. Therefore, we suggest a
multilevel algorithm that effectively finds a near-optimal shape parameter, which helps
to significantly reduce the error. Comparison to the finite element method and to the
generalised finited if ferences obtained in the flatlimits of the Gaussian RBF is provided.

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