Mimetic finite difference method for the stokes problem on polygonal meshes

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Various approaches to extend the finite element methods to non-traditional elements (pyramids, polyhedra, etc.) have been developed over the last decade. Building of basis functions for such elements is a challenging task and may require extensive geometry analysis. The mimetic finite difference (MFD) method has many similarities with low-order finite element methods. Both methods try to preserve fundamental properties of physical and mathematical models. The essential difference is that the MFD method uses only the surface representation of discrete unknowns to build stiffness and mass matrices. Since no extension inside the mesh element is required, practical implementation of the MFD ... continued below

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Lipnikov, K; Beirao Da Veiga, L; Gyrya, V & Manzini, G January 1, 2009.

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Various approaches to extend the finite element methods to non-traditional elements (pyramids, polyhedra, etc.) have been developed over the last decade. Building of basis functions for such elements is a challenging task and may require extensive geometry analysis. The mimetic finite difference (MFD) method has many similarities with low-order finite element methods. Both methods try to preserve fundamental properties of physical and mathematical models. The essential difference is that the MFD method uses only the surface representation of discrete unknowns to build stiffness and mass matrices. Since no extension inside the mesh element is required, practical implementation of the MFD method is simple for polygonal meshes that may include degenerate and non-convex elements. In this article, we develop a MFD method for the Stokes problem on arbitrary polygonal meshes. The method is constructed for tensor coefficients, which will allow to apply it to the linear elasticity problem. The numerical experiments show the second-order convergence for the velocity variable and the first-order for the pressure.

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  • Journal Name: Journal of Computational Physics

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  • Report No.: LA-UR-09-00753
  • Report No.: LA-UR-09-753
  • Grant Number: AC52-06NA25396
  • Office of Scientific & Technical Information Report Number: 956391
  • Archival Resource Key: ark:/67531/metadc930059

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  • January 1, 2009

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  • Nov. 13, 2016, 7:26 p.m.

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  • Dec. 12, 2016, 6:05 p.m.

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Lipnikov, K; Beirao Da Veiga, L; Gyrya, V & Manzini, G. Mimetic finite difference method for the stokes problem on polygonal meshes, article, January 1, 2009; [New Mexico]. (digital.library.unt.edu/ark:/67531/metadc930059/: accessed October 15, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.