Abstract
In the construction of nine point scheme, both vertex unknowns and cell-centered unknowns are introduced, and the vertex unknowns are usually eliminated by using the interpolation of neighboring cell-centered unknowns, which often leads to lose accuracy. Instead of using interpolation, here we propose a different method of calculating the vertex unknowns of nine point scheme, which are solved independently on a new generated mesh. This new mesh is a Voronoï mesh based on the vertexes of primary mesh and some additional points on the interface. The advantage of this method is that it is particularly suitable for solving diffusion problems with discontinuous coefficients on highly distorted meshes, and it leads to a symmetric positive definite matrix. We prove that the method has first-order convergence on distorted meshes. Numerical experiments show that the method obtains nearly second-order accuracy on distorted meshes.
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This work was partially supported by the National Basic Research Program (Grant No. 2005CB321703), the National Nature Science Foundation of China (Grant No. 90718029), and the Basic Research Project of National Defense (Grant No. A1520070074)
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Yuan, G., Sheng, Z. Calculating the vertex unknowns of nine point scheme on quadrilateral meshes for diffusion equation. Sci. China Ser. A-Math. 51, 1522–1536 (2008). https://doi.org/10.1007/s11425-008-0108-x
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DOI: https://doi.org/10.1007/s11425-008-0108-x