C2-Element radial basis function methods for some continuum mechanics problems

This work attempts to contribute further knowledge to high-order approximation and associated advanced echniques/methods for the numerical solution of differential equations in the discipline of computational science and engineering. Of particular interest is the numerical simulation of heat conduction, highly non-linear flows and multiscale problems. The distinguishing feature in this study is the development of novel local compact 2-node integrated radial basis function elements (IRBFEs) and their incorporation into the subregion/point collocation formulations based on Cartesian grids. As a result, a new class of C2-continuous methods are devised, representing a significant improvement on the usual C0-continuous methods. Incorporation of the new C2-continuous methods into the development of a high-order multiscale computational framework provides advantageous features compared to other multiscale frameworks available in the literature, including (i) high rates of convergence and levels of accuracy; and (ii) converged C2-continuous solutions of two-dimensional multiscale elliptic problems. Firstly, a new control-volume (CV) discretisation method, based on Cartesian grid and IRBFEs, for solving PDEs is proposed. Unlike the standard CV method (Patankar 1980), the flux values at CV faces are presently estimated with high-order IRBF approximations on 2-node elements and the solution is C2-continuous across the interface between two adjacent elements. Only two RBF centres (a smallest RBF set) associated with the two nodes of the element are used to construct the approximations locally leading to a very sparse and banded system matrix. Moreover, a wide range of RBF-widths can be used to effectively control the solution accuracy. Secondly, the proposed 2-node IRBFEs are incorporated into the subregion and point collocation frameworks for the discretisation of the streamfunction-vorticity formulation governing the fluid flows. Several high-order upwind schemes based on 2-node IRBFEs are developed for highly non-linear flows. Thirdly, the ADI procedure (Peaceman and Rachford 1955, Douglas and Gunn 1964) is applied to enhance the efficiency of the proposed methods. Especially novel C2-continuous compact schemes based on 2-node IRBFEs are devised and combined with the ADI procedure to yield optimal tridiagonal system matrices on each and every grid line. Such tridiagonal matrices can be solved effectively and efficiently with the Thomas algorithm (Fletcher 1991, Pozrikidis 1997). Finally, the proposed C2-continuous CV method is employed in a multiscale basis function approach to develop a high-order multiscale CV method for the solution of multiscale elliptic problems. Accuracy, stability and efficiency of the proposed methods are verified with extensive numerical results.