A SECOND-ORDER FLUX APPROXIMATION
FOR THE MIMETIC FINITE DIFFERENCE APPROXIMATION
OF DIFFUSION PROBLEMS
L. Beirao da
1
Veiga ,
K.
2
Lipnikov ,
G.
3
Manzini
Dipartimento di Matematica “G. Enriques”, Università degli Studi di Milano, Italy
2 Los Alamos National Laboratory, Theoretical Division, Los Alamos, New Mexico, US
3 Istituto di Matematica Applicata e Tecnologie Informatiche - CNR, Pavia, Italy
1
INTRODUCTION. The Mimetic Finite Difference (MFD) method is designed to mimic
HIGHER-ORDER FLUX APPROXIMATION
essential properties of the PDEs and the fundamental identities of the vector and tensor
calculus. The MFD method provides a low-order discretization of the diffusion equation
in mixed form on general polyhedral meshes. The numerical scheme is second-order
accurate for the scalar variable due to a superconvergence effect, but the flux
approximation is only first-order accurate. In this work, we present a high-order extension
of the MFD scheme, which is second-order accurate for both the scalar and the flux
approximation.
DESCRIPTION.
NUMERICAL RESULTS: a sequence of “randomized” grids,
where refined grids are not nested into coarser meshes.
THEORETICAL RESULTS:
(circles)
(squares)
P1 local consistency with constant diffusion tensor:
Modified P1 local consistency with non-constant diffusion tensor:
REFERENCES
1) L. Beirão da Veiga, K. Lipnikov, and G. Manzini. Convergence analysis of the high-order mimetic finite difference method. (2008). To appear in Numerische Mathematik.
2) L. Beirão da Veiga and G. Manzini. A higher-order formulation of the mimetic finite difference method. SIAM, J. Sci. Comput., 31(1):732-760, 2008.