A C2-continuous control-volume technique based on cartesian grids and two-node integrated-RBF elements for second-order elliptic problems

Abstract

This paper presents a new control-volume discretisation method, based on Cartesian grids and integrated-radial-basis-function elements (IRBFEs), for the solution of second-order elliptic problems in one and two dimensions. The governing equation is discretised by means of the control-volume formulation and the division of the problem domain into non-overlapping control volumes is based on a Cartesian grid. Salient features of the present method include (i) an element is defined by two adjacent nodes on a grid line, (ii) the IRBF approximations on each element are constructed using only two RBF centres (a smallest RBF set) associated with the two nodes of the element and (iii) the IRBFE solution is C2-continuous across the interface between two adjacent elements. The first feature guarantees consistency of the flux at control-volume faces. The second feature helps represent curved profiles between 2 adjacent nodes and leads to a sparse and banded system matrix, facilitating the employment of a large number of nodes. The third feature enhances the smoothness of element-based solutions, allowing a better estimation for the physical quantities involving derivatives. Numerical results indicate that (i) the proposed method can work with a wide range of the shapeparameter/RBF-width and (ii) the proposed technique yields more accurate results and faster convergence, especially for the approximation of derivatives, than the standard control-volume technique.