Meshless Radial Basis Function Collocation Methods for Interface Problems

Abstract

Numerical analysts paid much attention to the challenging PDEs defined on irregular geometries with fixed or moving interfaces. Such PDE models carry importance in many areas of sciences and engineering. These PDE models are challenging both from modelling and from solution point of view. Analytical solutions of such PDEs are rare or even do not exist. Therefore, simple, accurate and efficient numerical algorithms along with high performance computing devices are required to obtain numerical solution of such models. Fortunately, both hardware and software (algorithms and programming languages) evolved with the time to find approximate solution of the real life PDE models. Due to this evolutionary process, different variants of numerical methods have been reported in the literature over the decades. Some of the prominent methods are finite difference methods, finite volume methods, finite element methods, spectral and meshless methods. In the present thesis, both global and local meshless collocation methods are proposed for numerical solution of two-dimensional elliptic and parabolic PDEs with closed interface boundaries. These interface PDEs are solved on regular and irregular geometries with regular and irregular interface settings. Interface PDEs arise in many practical problems modelled on multiply connected domains, embodying various forms of discontinuities in the solution and/or its derivatives. The current work extends applications of the conventional Kansa method and the local meshless differential quadrature method (both in conventional and integrated forms) to numerical solution of the interface PDE models. Numerical evidence would reveal accurate performance of the meshless methods versus conventional methods, for different types of interface PDEs. Multiquadric (MQ) radial basis functions (RBFs) and its integrated form are used to construct global collocation methods for numerical solution of two-dimensional ellip tic/parabolic PDEs with curved or closed interface. The main purpose of this work is to perform a comparative analysis of both the methods via accuracy and condition number of the coefficient matrices for elliptic interface PDEs. A set of scattered nodes (Halton points) is considered on both sides of the interface of the domain of the benchmark interface PDEs. The major addition of the current work is the successful implementation of the local version of the meshless collocation methods for numerical solution of two-dimensional in terface PDEs having interfaces with sharp corners and the Stokes problem. The innovative mechanism of node placing algorithm based on the electrostatic node repulsion procedure [1] is used to generate interior nodes in each subdomain. Placements of the uniform nodes are ensured on each segment of the outer boundary and the interface to avoid nodes clus tering at the sharp corners. Similarly, the local meshless method based on MQ RBF is also used for numerical solution of Stokes problem. The Stokes equations are collocated in a rectangular region with different interface conditions and direct solvers are used instead of iterative algorithm for the velocity and pressure equations. Due to the unknown boundary conditions for the pressure term in the Stokes equations, value of the pressure is obtained by integrating the Stokes equations and eliminating the constant of integration.