A Comparison Between Meshless Radial Basis Function Collocation Method and Finite Element Method for Solving Poisson and Stokes Problems

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 conducted using Backward Euler Method. The performances are assessed over the accuracy, runtime, condition number, and ease of implementation criteria. Three kinds of errors were calculated; least square error, root mean square error and maximum relative error. To calculate the least square error while using meshless Radial Basis Function Collocation Method, a novel technique is implemented. Imaginary numerical solution surfaces are created and then the volume between those imaginary surfaces and the analytic solution surfaces is calculated, enabling a fair error calculation. Lastly, all solutions are put together and solution trends are observed over the number of solution nodes vs. runtime, accuracy vs. runtime, and accuracy vs. the number of nodes. The assessment indicates the criteria under which Finite Element Method perform better and those when Radial Basis Function Collocation Method outperforms its mesh dependent counterpart.