Convergence of Grid Boundary-Value Problems for Functions Defined on Grid Cells and Faces

  • N. V. ArdelyanEmail author
  • K. V. KosmachevskiiEmail author


For stationary diffusion-type equations, we study the convergence of grid inhomogeneous boundary-value problems of a version of the mimetic finite difference (MFD) technique in which grid scalars are defined inside grid cells and grid vectors are specified by their local normal coordinates on the plane faces of grid cells. Grid equations and boundary conditions are formulated in operator form using consistent grid analogs of invariant first-order differential operators and of boundary operators. Convergence is studied on the basis of the theory of operator difference schemes; i.e., a priori estimates for the norm of the solution error in terms of the norm of the approximation error are obtained that guarantee convergence of the first order under inhomogeneous boundary conditions of the first, second, and third kind in a domain with a curvilinear boundary. Grid analogs of embedding inequalities and approximation relations obtained earlier are used.


grid operators polyhedral grid cells faces approximation error a priori estimates convergence 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. A. Samarskii, The Theory of Difference Schemes (Nauka, Moscow, 1977; CRC, Boca Raton, FL, 2001).zbMATHGoogle Scholar
  2. 2.
    A. A. Samarskii and V. B. Andreev, Difference Methods for Elliptic Equations (Nauka, Moscow, 1976) [in Russian].zbMATHGoogle Scholar
  3. 3.
    K. N. Lipnikov, G. Manzini, and M. J. Shashkov, “Mimetic finite difference method,” J. Comput. Phys., Part B 257, 1163–1227 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    N. V. Ardelyan, K. V. Kosmachevskii, and M. N. Sablin, “Properties of consistent grid operators for grid functions defined inside grid cells and on grid faces,” Comput. Math. Model. 29, 10–29 (2018).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    N. V. Ardelyan, K. V. Kosmachevskii, and M. N. Sablin, “Properties of grid boundary value problems for functions defined on grid cells and faces,” Moscow Univ. Comput. Math. Cybern. 41, 105–112 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    M. N. Sablin, N. V. Ardelyan, and K. V. Kosmachevskii, “Consistent grid operators with the cell-nodal definition of grid functions,” Moscow Univ. Comput. Math. Cybern. 41, 1–10 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    N. V. Ardelyan and I. S. Gushchin, “One approach to the construction of completely conservative difference schemes,” Moscow Univ. Comput. Math. Cybern. 3, 1–9 (1982).zbMATHGoogle Scholar
  8. 8.
    N. V. Ardelyan, “The convergence of difference schemes for two-dimensional equations of acoustics and maxwell’s equations,” U. S. S. R. Comput. Math. Math. Phys. 23(5), 93–99 (1983).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    N. V. Ardelyan and K. V. Kosmachevskij, “Implicit free-lagrange method for computing two-dimensional magnetogas-dynamic flows,” Comput. Math. Model. 6, 209–224 (1995).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    N. V. Ardeljan and K. V. Kosmachevskii, “An implicit free Lagrange method with finite element operators for the solution of MHD-problems,” in Finite Elements in Fluids, New Trends and Applications, Proceedings of the IACM Special International Conference, Venezia, Italy, 1995, Part 2, pp. 1099–1108.Google Scholar
  11. 11.
    N. V. Ardelyan and S. V. Chernigovskij, “Stability of an operator-difference scheme defined on a direct sum of spaces,” Moscow Univ. Comput. Math. Cybern. 1, 41–47 (1984).zbMATHGoogle Scholar
  12. 12.
    N. V. Ardelyan, “Stability of two-layer operator-difference schemes with symmetric and skew-symmetric operators,” Fundam. Prikl. Mat. 5(4), 79–91 (1999).MathSciNetzbMATHGoogle Scholar

Copyright information

© Allerton Press, Inc. 2019

Authors and Affiliations

  1. 1.Department of Computational Mathematics and CyberneticsMoscow State UniversityMoscowRussia

Personalised recommendations