Performance analyses of mesh-based local Finite Element Method and meshless global RBF Collocation Method for solving Poisson and Stokes equations

Abstract

Steady and unsteady Poisson and Stokes equations are solved using mesh dependent Finite Element Method and meshless Radial Basis Function Collocation Method to compare the performances of these two numerical techniques across several criteria. The accuracy of Radial Basis Function Collocation Method with multiquadrics is enhanced by implementing a shape parameter optimization algorithm. For the time-dependent problems, time discretization is done using Backward Euler Method. The performances are assessed over the accuracy, runtime, condition number, and ease of implementation. Three error kinds considered; least square error, root mean square error and maximum relative error. To calculate the least square error using meshless Radial Basis Function Collocation Method, a novel technique is implemented. Imaginary numerical solution surfaces are created, then the volume between those imaginary surfaces and the analytic solution surfaces is calculated, ensuring a fair error calculation. Lastly, all results are put together and trends are observed. The change in runtime vs. accuracy and number of nodes; and the change in accuracy vs. the number of nodes is analyzed. The study indicates the criteria under which Finite Element Method performs better and conditions when Radial Basis Function Collocation Method outperforms its mesh dependent counterpart.