The cell functional minimization scheme for the anisotropic diffusion problems on arbitrary polygonal grids∗
Revised: 25 March 2014
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 H1 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.
Mathematics Subject Classification: 65N12 / 65N08 / 35J25
Key words: Cell functional minimization / finite volume scheme / diffusion problem / polygonal mesh / convergence / stability / error estimate
This work of the first author is supported by the National Natural Science Foundation of China under contract No. 91118001 and the Science Foundation of China Academy of Engineering Physics (2013B0202034), the second author is sponsored by the National Natural Science Foundation of China under contract Nos. 91330205 and 11271053, and the corresponding author is under the auspices of the National Natural Science Foundation of China (91330107) and Foundation of President of China Academy of Engineering Physics (2014-1-042).
© EDP Sciences, SMAI 2015