Research activity at p2logo and p2logo

This page illustrate the common effort between Pavia and Los Alamos in developing the technology of the Mimetic Finite Difference methods.

The research activity on mimetic schemes at IMATI-CNR has been developed out of a collaboration starting in 2005 with the team of researchers working on mimetic schemes in the Mathematical Modeling and Analysis Group (T5) of the Los Alamos National Laboratory in Los Alamos, New Mexico.

This initial collaboration produced the seminal paper authored by Brezzi, Lipnikov, Shashkov [13] and published in 2005 on SINUM.

Since then, other italian researchers working in IMATI and in places other than Pavia have joined this project.

We list below (almost in chronological order) the main topics that have been considered so far:

  • a mimetic formulation and related theoretical analysis for meshes of arbitrary types of elements e.g., tetrahedrons, pyramids, hexahedrons, degenerate polyhedrons, etc., cf. [13], [14];

  • a mimetic formulation and related theoretical analysis for meshes with curved-faced elements, cf. [15], [16];

  • a post-processing technique for the scalar solution based on an inexpensive element-wise flux reconstruction, cf. [18];

  • an a-posteriori indicator and its application to mesh adaptivity, cf. [1], [6];

  • a higher-order mimetic approximation of the numerical fluxes for diffusion problems, cf. [7], [8];

  • a new mimetic formulation for diffusion problems based on nodal unknowns, cf. [12];

  • an extension to convection-diffusion problems, cf. [19], and a unified formulation with finite volume methods (also called Hybrid Mixed Method), [3]

  • a new mimetic formulation for the steady Stokes equation in two dimensions, cf. [4], and in three dimensions, cf.[9], and with a reduced bubble stabilization [5];

  • a new mimetic formulation for linear elasticity, cf. [2];

  • a mimetic formulation for electromagnetic problems, cf. [11], [20]

  • a mimetic formulation for eigenvalue problems in mixed form, cf. [17], a mimetic modification to standard mixed FEM on quadrilaterals to improve the convergence behavior, cf. [10].

Bibliography

  1. L. Beirão da Veiga. A residual based error estimator for the Mimetic Finite Difference method. Numerische Mathematik, 108:387-406, 2008.

  2. L. Beirão da Veiga. A mimetic discretization method for linear elasticity. ESAIM: Mathematical Modeling and Numerical Analysis , 44(2):231-250, 2010

  3. L. Beirão da Veiga, J. Droniou, and G. Manzini. A unified approach to handle convection term in Finite Volumes and Mimetic Discretization Methods for elliptic problems. Technical Report 23PV09/19/0, IMATI-CNR (2009). Accepted for publication by IMA Journal on Numerical Analysis.

  4. L. Beirão da Veiga, V. Gyrya, K. Lipnikov, and G. Manzini. Mimetic finite difference method for the Stokes problem on poligonal meshes. Journal of Computational Physics, 228(19):7215-7232, 2009.

  5. L. Beirão da Veiga, and K. Lipnikov. A mimetic discretization for the Stokes problem with selected edge bubbles. SIAM, J. Sci. Comput., 32(2):875-893, 2010.

  6. L. Beirão da Veiga and G. Manzini. An a-posteriori error estimator for the mimetic finite difference approximation of elliptic problems. International Journal of Numerical Methods in Engineering, 76(11):1696-1723, 2008.

  7. 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.

  8. L. Beirão da Veiga, K. Lipnikov, and G. Manzini. Convergence analysis of the high-order mimetic finite difference method. Numerische Mathematik, 113(3): 325—356, 2009.

  9. L. Beirão da Veiga, K. Lipnikov, and G. Manzini. The Mimetic Finite Difference method for steady Stokes problems on polyhedral meshes. Technical Report 6PV09/5/0, IMATI-CNR (2009). Accepted for publication to SIAM Journal on Numerical Analysis.

  10. D. Boffi and L. Gastaldi. Some remarks on quadrilateral mixed finite elements. Computers and Structures, 87(11-12):751-757, 2009.

  11. F. Brezzi, A. Buffa. Innovative mimetic discretizations for electromagnetic problems. To appear in Journal of Computational and Applied Mathematic.

  12. F. Brezzi, A. Buffa, K. Lipnikov. Mimetic finite differences for elliptic problems. ESAIM: Mathematical Modeling and Numerical Analysis, 43:277-295, 2009.

  13. F. Brezzi, K. Lipnikov, M. Shashkov. Convergence of mimetic finite difference methods for diffusion problems on polyhedral meshes. SIAM Journal on Numerical Analysis, 43(5):1872-1896, 2005.

  14. F. Brezzi, K. Lipnikov, V. Simoncini. A family of mimetic finite difference methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci. 15(10):1533-1553, 2005.

  15. F. Brezzi, K. Lipnikov, M. Shashkov. Convergence of mimetic finite difference methods for diffusion problems on polyhedral meshes with curved faces. Math. Models Methods Appl. Sci. 16(2):275-298, 2006.

  16. F. Brezzi, K. Lipnikov, M. Shashkov, V. Simoncini. A new discretization methodology for diffusion problems on generalized polyhedral meshes. Computer Methods in Applied Mechanics and Engineering. 196:3682-3692, 2007.

  17. A. Cangiani, F. Gardini, and G. Manzini. Convergence of the mimetic finite difference method for eigenvalue problems in mixed form . Computer Methods in Applied Mechanics and Engineering. In press, available online since 12 June 2010. Also Technical Report 31PV09/24/0, IMATI-CNR (2009).

  18. A. Cangiani and G. Manzini. Flux reconstruction and solution post-processing in mimetic finite difference methods. Computer Methods in Applied Mechanics and Engineering, 197(9-12):933-945, 2008.

  19. A. Cangiani, G. Manzini, and A. Russo. Convergence analysis of a mimetic finite difference method for general second-order elliptic problems. SIAM Journal on Numerical Analysis, 47(4): 2612-2637, 2009.

  20. K. Lipnikov, G. Manzini, F. Brezzi, A. Buffa. The mimetic finite difference method for 3D magnetostatics fields problems. Technical Report 30PV09/23/0, IMATI-CNR (2009). Journal of Computational Physics. Accepted for publication (September 2010).