Infinitely smooth multiquadric RBFs combined high-resolution compact discretization for nonlinear 2D elliptic PDEs on a scattered grid network

Multiquadric radial basis functions combined with compact discretization to estimate solutions of two dimensions nonlinear elliptic type partial differential equations are presented. The scattered grid network with continuously varying step sizes helps tune the solution accuracies depending upon the location of high oscillation. The radial basis functions employing a nine-point grid network are used to improve the functional evaluations by compact formulation, and it saves memory space and computing time. A detailed description of convergence theory is presented to estimate the error bounds. The analysis is based on a strongly connected graph of the Jacobian matrix, and their monotonicity occurred in the scheme. It is shown that the present strategy improves the approximate solution values for the elliptic equations exhibiting a sharp changing character in a thin zone. Numerical simulations for the convection-diffusion equation, Graetz-Nusselt equation, Schr¨odinger equation, Burgers equation, and Gelfand-Bratu equation are reported to illustrate the utility of the new algorithm.