Various approaches to extend finite element methods to non-traditional elements (general polygons, pyramids, polyhedra, etc.) have been developed over the last decade. The construction of basis functions for such elements is a challenging task and may require extensive geometrical analysis. The mimetic finite difference (MFD) method works on general polygonal meshes and has many similarities with low-order finite element methods. Both schemes try to preserve the fundamental properties of the underlying physical and mathematical models. The essential difference between the two schemes is that the MFD method uses only the surface representation of discrete unknowns to build the stiffness and mass matrices. Since no extension of basis functions inside the mesh elements 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 present a new MFD method for the Stokes problem on arbitrary polygonal meshes and analyze its stability. The method is developed for the general case of tensor coefficients, which allows us to apply it to a linear elasticity problem, as well. Numerical experiments show, for the velocity variable, second-order convergence in a discrete L2 norm and first-order convergence in a discrete H1 norm. For the pressure variable, first-order convergence is shown in the L2 norm. © 2009 Elsevier Inc.

BEIRAO DA VEIGA, L., Gyrya, V., Lipnikov, K., Manzini, G. (2009). Mimetic finite difference method for the Stokes problem on polygonal meshes. JOURNAL OF COMPUTATIONAL PHYSICS, 228(19), 7215-7232 [10.1016/j.jcp.2009.06.034].

### Mimetic finite difference method for the Stokes problem on polygonal meshes

#### Abstract

Various approaches to extend finite element methods to non-traditional elements (general polygons, pyramids, polyhedra, etc.) have been developed over the last decade. The construction of basis functions for such elements is a challenging task and may require extensive geometrical analysis. The mimetic finite difference (MFD) method works on general polygonal meshes and has many similarities with low-order finite element methods. Both schemes try to preserve the fundamental properties of the underlying physical and mathematical models. The essential difference between the two schemes is that the MFD method uses only the surface representation of discrete unknowns to build the stiffness and mass matrices. Since no extension of basis functions inside the mesh elements 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 present a new MFD method for the Stokes problem on arbitrary polygonal meshes and analyze its stability. The method is developed for the general case of tensor coefficients, which allows us to apply it to a linear elasticity problem, as well. Numerical experiments show, for the velocity variable, second-order convergence in a discrete L2 norm and first-order convergence in a discrete H1 norm. For the pressure variable, first-order convergence is shown in the L2 norm. © 2009 Elsevier Inc.
##### Scheda breve Scheda completa Scheda completa (DC)
Articolo in rivista - Articolo scientifico
Incompressible Stokes equations; Mimetic discretization; Polygonal mesh
English
2009
228
19
7215
7232
none
BEIRAO DA VEIGA, L., Gyrya, V., Lipnikov, K., Manzini, G. (2009). Mimetic finite difference method for the Stokes problem on polygonal meshes. JOURNAL OF COMPUTATIONAL PHYSICS, 228(19), 7215-7232 [10.1016/j.jcp.2009.06.034].
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/10281/99166`
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