Improving Volume Element Methods by Meshless Radial Basis Function Techniques
P. Orsini; H. Power; and H. Morvan

doi:10.3970/cmes.2008.023.187
Source CMES: Computer Modeling in Engineering & Sciences, Vol. 23, No. 2, pp. 187-208, 2008
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Abstract In this work, we present a modified Control Volume (CV) method that uses a Radial Basis Function (RBF) interpolation to improve the prediction of the flux accuracy at the faces of the CV. The method proposed differs from classical CV methods in the way that the flux at the cell surfaces is computed. A local RBF interpolation of the field variable is performed at the centres of the cell being integrated and its neighbours. This interpolation is then used to reconstruct the solution and its gradient in the integration points which support the flux computation. In addition, it is required that such interpolation satisfies the governing equation in a certain number of points placed around the cell centres. In this way, the local interpolations become equivalent to local boundary-value problems. To find the solution to the local problems, we have tested both the unsymmetric (Kansa's method) and symmetric (Hermitian method) RBF approaches. The proposed CV approach will be referred to as the CV-RBF method and validated here in a series of one- and three-dimensional test cases.
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