Geodesic
The cell functional minimization scheme for the anisotropic diffusion problems on arbitrary polygonal grids
Yin, Li1 ; Wu, Jiming1 ; Gao, Zhiming1
1 Institute of Applied Physics and Computational Mathematics, P.O. Box 8009-9, Beijing 100088, P.R. China.
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 1, pp. 193-220

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A finite volume scheme based on minimization of a certain cell functional is constructed for unstructured polygonal meshes. This new scheme has a local stencil, allows arbitrary diffusion tensors, leads to a symmetric positive definite diffusion matrix in case that edge unknowns are defined at the midpoints of edges, and is linearity-preserving, i.e., preserves linear solutions. Under a very weak geometry condition, the stability result and discrete H 1 error estimate of the scheme is obtained through a discrete functional approach. Finally, numerical results on various mesh types (including a particular jigsaw puzzle mesh) demonstrate the good performance of the scheme and validate the theoretical analysis.

DOI : 10.1051/m2an/2014030
Classification : 65N12, 65N08, 35J25
Keywords: Cell functional minimization, finite volume scheme, diffusion problem, polygonal mesh, convergence, stability, error estimate

Yin, Li 1 ; Wu, Jiming 1 ; Gao, Zhiming 1

1 Institute of Applied Physics and Computational Mathematics, P.O. Box 8009-9, Beijing 100088, P.R. China. 
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     title = {The cell functional minimization scheme for the anisotropic diffusion problems on arbitrary polygonal grids},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {193--220},
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     volume = {49},
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Yin, Li; Wu, Jiming; Gao, Zhiming. The cell functional minimization scheme for the anisotropic diffusion problems on arbitrary polygonal grids. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 1, pp. 193-220. doi: 10.1051/m2an/2014030

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