Edward J. Kansa

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

This study examines the generalized multiquadrics (MQ), @fj(x) = [(x-xj)^2+cj^2]^@b in the numerical solutions of elliptic two-dimensional partial differential equations (PDEs) with Dirichlet boundary conditions. The exponent @b as well as cj^2 can be classified as shape parameters since these affect the shape of the MQ basis function. We examined variations of @b as well as cj^2 where cj^2 can be different over the interior and on the boundary. The results show that increasing ,@b has the most important effect on convergence, followed next by distinct sets of (cj^2)@W@?@W @? (cj^2)@?@W. Additional convergence accelerations were obtained by permitting both (cj^2)@W@?@W and (cj^2)@?@W 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.

Collocation And Optimization Initialization

Integrated volumetric methods such as finite elements and their “meshless” variations are typically smoother than the strong form finite difference and radial basis function collocation methods. Numerical methods decrease their convergence rates with successively higher orders of differentiation along with improved conditioning. In contrast, increasing order of integration increases the convergence rate at the expense of poorer conditioning. In the study presented, a two-dimensional Poisson equation with exponential dependency is solved. The solution of the point collocation problem becomes the initial estimate for an integrated volumetric minimization process. Global, rather than local integration, is used since there is no need to construct any meshes for integration as done in the “meshless” finite element analogs. The root mean square (RMS) errors are compared. By pushing the shape parameter to very large values, using extended precision, the RMS errors show that spatial refinement benefits are relatively small compared to pushing shape parameters to increasing larger values. The improved Greedy Algorithm was used to optimize the set of data and evaluation centers for various shape parameters. Finally, extended arithmetic precision is used to push the range of the shape parameters.