The role of the multiquadric shape parameters in solving elliptic partial differential equations

This study examines the generalized multiquadrics (MQ), φ_j(x) = [(x-x_j)²+c_j ²] ^β in the numerical solutions of elliptic two-dimensional partial differential equations (PDEs) with Dirichlet boundary conditions. The exponent β as well as c_j ² can be classified as shape parameters since these affect the shape of the MQ basis function. We examined variations of β as well as c_j ² where c_j ² can be different over the interior and on the boundary. The results show that increasing,β has the most important effect on convergence, followed next by distinct sets of (c_j ²)Ω∂Ω ≪ (c_j ²)∂Ω. Additional convergence accelerations were obtained by permitting both (c_j ²)Ω∂Ω and (c_j ²)∂Ω to oscillate about its mean value with amplitude of approximately 1/2 for odd and even values of the indices. Our results show high orders of accuracy as the number of data centers increases with some simple heuristics.