Geodesic
A priori error estimates of the local discontinuous Galerkin method on staggered grids
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 24 (2020) no. 1, pp. 116-136.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we present a priori error analysis of the solution of a homogeneous boundary value problem for a second-order differential equation by the Discontinuous Galerkin method on staggered grids. The spatial discretization is constructed using an appeal to a mixed finite element formulation. Second-order derivatives cannot be directly matched in a weak variational formulation using the space of discontinuous functions. For lower the order, the components of the flow vector are considered as auxiliary variables of the desired second-order equation. The approximation is based on staggered grids. The main grid consists of triangles, the dual grid consists of median control volumes around the nodes of the triangular grid. The approximation of the desired function is built on the cells of the main grid, while the approximation of auxiliary variables is built on the cells of the dual grid. To calculate the flows at the boundary between the elements, a stabilizing parameter is used. Moreover, the flow of the desired function does not depend on auxiliary functions, while the flow of auxiliary variables depends on the desired function. To solve this problem, the necessary lemmas are formulated and proved. As a result, the main theorem is formulated and proved, the result of which is a priori estimates for solving a parabolic equation using the discontinuous Galerkin method. The main role in the analysis of convergence is played by the estimate for the negative norm of the gradient. We show that for stabilization parameter of first order, the $L^2$-norm of the solution is of order $k+{1}/{2}$, if stabilization parameter of order $h^{-1}$ is taken, the order of convergence of the solution increases to $k+1$, when polynomials of total degree at least $k$ are used.
Keywords: a priori error analysis, finite element method, discontinuous Galerkin method, staggered grids, parabolic problems. 
@article{VSGTU_2020_24_1_a6,
     author = {R. V. Zhalnin and V. F. Masyagin and E. E. Peskova and V. F. Tishkin},
     title = {A priori error estimates of the local discontinuous {Galerkin} method on staggered grids},
     journal = {Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences},
     pages = {116--136},
     publisher = {mathdoc},
     volume = {24},
     number = {1},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSGTU_2020_24_1_a6/}
}
TY  - JOUR
AU  - R. V. Zhalnin
AU  - V. F. Masyagin
AU  - E. E. Peskova
AU  - V. F. Tishkin
TI  - A priori error estimates of the local discontinuous Galerkin method on staggered grids
JO  - Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences
PY  - 2020
SP  - 116
EP  - 136
VL  - 24
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/VSGTU_2020_24_1_a6/
LA  - ru
ID  - VSGTU_2020_24_1_a6
ER  - 
%0 Journal Article
%A R. V. Zhalnin
%A V. F. Masyagin
%A E. E. Peskova
%A V. F. Tishkin
%T A priori error estimates of the local discontinuous Galerkin method on staggered grids
%J Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences
%D 2020
%P 116-136
%V 24
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/VSGTU_2020_24_1_a6/
%G ru
%F VSGTU_2020_24_1_a6
R. V. Zhalnin; V. F. Masyagin; E. E. Peskova; V. F. Tishkin. A priori error estimates of the local discontinuous Galerkin method on staggered grids. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 24 (2020) no. 1, pp. 116-136. http://geodesic.mathdoc.fr/item/VSGTU_2020_24_1_a6/

[1] Masyagin V. F., Zhalnin R. V., Tishkin V. F., “Discontinuous finite-element Galerkin method for numerical solution of two-dimensional diffusion problems on unstructured grids”, Zhurnal SVMO, 15:2 (2013), 59–65 (In Russian) | Zbl

[2] Zhalnin R. V., Ladonkina M. E., Masyagin V. F., Tishkin V. F., “Discontinuous finite-element Galerkin method for numerical solution of two-dimensional diffusion problems on unstructured grids”, Zhurnal SVMO, 16:2 (2014), 7–13 (In Russian) | Zbl

[3] Zhalnin R. V., Ladonkina M. E., Masyagin V. F., Tishkin V. F., “Solution of 3D heat conduction equations using the discontinuous Galerkin method on unstructured grids”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 19:3 (2015), 523–533 (In Russian) | DOI | Zbl

[4] Zhalnin R. V., Ladonkina M. E., Masyagin V. F., Tishkin V. F., “Solving the problem of non-stationary filtration of substance by the discontinuous Galerkin method on unstructured grids”, Comput. Math. Math. Phys., 56:6 (2016), 977–986 | DOI | DOI | Zbl

[5] Zhalnin R. V., Ladonkina M. E., Masyagin V. F., Tishkin V. F., “Discontinuous finite-element Galerkin method for numerical solution of parabolic problems in anisotropic media on triangle grids”, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 9:3 (2016), 144–151 (In Russian) | DOI | Zbl

[6] Zhalnin R. V., Masyagin V. F., “Galerkin method with discontinuous basis functions on staggered grips a priory estimates for the homogeneous Dirichlet problem”, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 11:2 (2018), 29–43 (In Russian) | DOI | Zbl

[7] Cockburn B., Shu C.-W., “The local discontinuous Galerkin finite element method for convection-diffusion systems”, SIAM J. Numer. Anal., 35:6 (1998), 2440–2463 | DOI | Zbl

[8] Bassi F., Rebay S., “A High-Order Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible Navier-Stokes Equations”, J. Comp. Phys., 131:2 (1997), 267–279 | DOI | Zbl

[9] Cockburn B., Hou S., Shu C.-W., “TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: The multidimensional case”, Math. Comp., 54:190 (1990), 545–581 | DOI | Zbl

[10] Cockburn B., Lin S.-Y., Shu C.-W., “TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One dimensional systems”, J. Comput. Phys., 84:1 (1989), 90–113 | DOI | Zbl

[11] Cockburn B., Shu C.-W., “TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework”, Math. Comp., 52:186 (1989), 411–435 | DOI | Zbl

[12] Cockburn B., Lin S.-Y., Shu C.-W., “The Runge–Kutta local projection $P^1$-discontinuous Galerkin method for scalar conservation laws”, ESAIM: Mathematical Modelling and Numerical Analysis, 25:3 (1991), 337–361 | DOI | Zbl

[13] Cockburn B., Shu C.-W., “The Runge–Kutta discontinuous Galerkin finite element method for conservation laws V: Multidimensional systems”, J. Comput. Phys., 141:2 (1998), 199–224 | DOI | Zbl

[14] Ciarlet P. G., The Finite Element Method for Elliptic Problems, Classics in Applied Mathematics, SIAM, Philadelphia, 2002, xxiii+529 pp. | DOI

[15] Castillo P., Cockburn B., Perugia I., Schötzau D., “An a priory error analysis of the local discontinuous Galerkin method for elliptic problems”, SIAM J. Numer. Anal., 38:5 (2000), 1676–1706 | DOI | Zbl

[16] Thomée V., Galerkin Finite Element Methods for Parabolic Problems, Springer Series in Computational Mathematics, 25, Springer, Berlin, 1997, x+302 pp. | DOI | Zbl

[17] Rivière B., Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations, Frontiers in Applied Mathematics, SIAM, Philadelphia, 2008, xxii+178 pp. | DOI | Zbl

[18] Pany A., Yadav S., “An $hp$-local discontinuous Galerkin method for parabolic integro-differential equations”, J. Sci. Comput., 46:1 (2011), 71–99 | DOI

[19] Babuška I., Suri M., “The $h-p$ version of the finite element method with quasiuniform meshes”, ESAIM: Mathematical Modelling and Numerical Analysis, 21:2 (1987), 199–238 | DOI | Zbl

[20] Dautov R. Z., Fedotov E. M., “Abstract theory of hybridizable discontinuous Galerkin methods for second-order quasilinear elliptic problems”, Comput. Math. Math. Phys., 54:3 (2014), 474–490 | DOI | DOI | Zbl