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Second-Order Flux Approximation for Mimetic Finite Difference in Diffusion Problems

This work extends the Mimetic Finite Difference (MFD) method to achieve second-order accuracy for both scalar variables and flux approximations in diffusion problems. The MFD method mimics key properties of PDEs, offering a low-order discretization on general polyhedral meshes. While classical MFD has first-order accuracy for flux approximations, our high-order extension successfully enhances accuracy. We present numerical results on randomized grids with theoretical consistency analysis, showcasing the improved performance of this new approach in computational applications.

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Second-Order Flux Approximation for Mimetic Finite Difference in Diffusion Problems

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  1. A SECOND-ORDER FLUX APPROXIMATIONFOR THE MIMETIC FINITE DIFFERENCE APPROXIMATIONOF DIFFUSION PROBLEMSL. Beirao da Veiga1, K. Lipnikov2, G. Manzini3 1 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 INTRODUCTION.The Mimetic Finite Difference (MFD) method is designed to mimic 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 accuratefor both the scalar and the flux approximation. HIGHER-ORDER FLUX APPROXIMATION DESCRIPTION. NUMERICAL RESULTS: a sequence of “randomized” grids, where refined grids are not nested into coarser meshes. THEORETICAL RESULTS: (circles) (squares) Modified P1 local consistency with non-constant diffusion tensor: P1 local consistency with constant diffusion tensor: REFERENCES 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. 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.

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