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

This study examines the generalized multiquadrics (MQ), φj(x) = [(x−xj)2+cj2]β in the numerical solutions of elliptic two-dimensional partial differential equations (PDEs) with Dirichlet boundary conditions. The exponent β as well as cj2 can be classified as shape parameters since these affect the shape of the MQ basis function. We examined variations of β as well as cj2 where cj2 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 (cj2)Ω∂Ω (cj2)∂Ω. Additional convergence accelerations were obtained by permitting both (cj2)Ω∂Ω and (cj2)∂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.