02. Zhalnin R. V., Masyagin V. F., Peskova Ye. Ye. A Priori Estimates of Soluting a Homogeneous Boundary Value Problem for Parabolic Type Equations by the Discontinuous Galerkin Method on Staggered Grids.

UDC 517.93

DOI: 10.15507/0236-2910.027.201704.490-503

A Priori Estimates of Solution of a Homogeneous Boundary Value Problem for Parabolic Type Equations by the Discontinuous Galerkin Method on Staggered Grids

Ruslan V. Zhalnin
Head of Chair of Applied Mathematics, Differential Equations and Theoretical Mechanics, National Research Mordovia State University (68 Bolshevistskaya St., Saransk 430005, Russia), Ph.D. (Physics and Mathematics), ORCID: http://orcid.org/0000-0002-1103-3321, This email address is being protected from spambots. You need JavaScript enabled to view it.

Viktor F. Masyagin
Associate Professor of Chair of Applied Mathematics, Differential Equations and Theoretical Mechanics, National Research Mordovia State University (68 Bolshevistskaya St., Saransk 430005, Russia), Ph.D. (Physics and Mathematics), ORCID: http://orcid.org/0000-0001-6738-8183, This email address is being protected from spambots. You need JavaScript enabled to view it.

Yelizaveta Ye. Peskova
Senior Lecturer of Chair of Applied Mathematics, Differential Equations and Theoretical Mechanics, National Research Mordovia State University (68 Bolshevistskaya St., Saransk 430005, Russia), This email address is being protected from spambots. You need JavaScript enabled to view it.

Introduction: 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.
Materials and Methods: This study is based on the unified hp-version error analysis of local discontinuous Galerkin method proposed by Castillo et al. [Optimal a priori error estimates for the hp-version of the local discontinuous Galerkin method for convection-diffusion problems, 2002]. The purpose of this paper is to present a new approach to the error analysis of the solution of parabolic equations by the discontinuous Galerkin method on staggered grids.
Results: We suggest that approximation errors depend on the characteristic size of the cells and the degree of polynomials used in the basis functions. The necessary lemmas are formulated for the problem solution. The complete proof of the lemmas formulated is carried out. We formulated and proved a theorem, in which a priori error estimates are given for solving parabolic equations using the discontinuous Galerkin method on staggered grids
Discussion and Conclusions: The obtained results are consistent with similar studies of other authors and complement them. Further work on this topic involves the study of diffusion-type equations of order higher than the first and the production of a posteriori error estimates.

Keywords: a priori error analysis, finite elements, discontinuous Galerkin methods, discontinuous basis functions, parabolic problems

For citation: Zhalnin R. V., Masyagin V. F., Peskova Ye. Ye. A Priori Estimates of Solution of a Homogeneous Boundary Value Problem for Parabolic Type Equations by the Discontinuous Galerkin Method on Staggered Grids. Vestnik Mordovskogo universiteta = Mordovia University Bulletin. 2017: 27(4):490–503. DOI: 10.15507/0236-2910.027.201704.490-503

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