Volume 9, no. 3Pages 144 - 151

Discontinuous Finite-Element Galerkin Method for Numerical Solution of Parabolic Problems in Anisotropic Media on Triangle Grids

R.V. Zhalnin, M.E. Ladonkina, V.F. Masyagin, V.F. Tishkin
A new numerical algorithm for solving parabolic initial-boundary values problems in anisotropic media is proposed. The algorithm is based on Galerkin method with discontinuous basic functions on triangle meshes. The 2nd order derivatives can't be directly harmonized in a weak variational formulation using the discontinuous functions' space. Hence additional variables are introduced to reduce the initial 2nd-order equation to the system of the 1st-order equations. The special feature of this method is in consideration of additional variables within a dual mesh. The dual mesh consists of median control values and is conjugate to the initial triangle mesh. The stream values on the element boundaries are calculated with addition of stabilizing additives. The method is studied basing on the example of 2-dimensional parabolic boundary problems. Convergence and accuracy of the method are investigated. Calculations in model problem show the possibility to use the method discussed for solving parabolic problems in anisotropic media on triangle meshes.
Full text
parabolic equations; anisotropic media; discontinuous Galerkin method; сonvergence and accuracy of the method.
1. Cockburn B. An Introduction to the Discontinuous Galerkin Method for Convection-Dominated Problems. Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, 1998, vol. 1697, pp. 151-268. DOI: 10.1007/BFb0096353
2. Fletcher C.A.J. Computational Galerkin Methods. N.Y., Springer, 1984.
3. Zhalnin R.V., Masyagin V.F., Panyushkina E.N. Discontinuous Galerkin Method for Numerical Solution of Twodimensional Diffusion Problems on Unstructural Stagerred Grids. Modern Problems of Science and Education, 2013, vol. 6. Available at: www.science-education.ru/113-10929
4. Ladonkina M.E., Neklyudova O.A., Tishkin V.F. Research of the Impact of Different Limiting Functions on the Order of Solution Obtained by RKDG. Mathematical Models and Computer Simulations, 2013, vol. 5, no. 4, pp. 346-349. DOI: 10.1134/S2070048213040091
5. Bassi F., Rebay S. A High-Order Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible Navier - Stokes Equations. Journal of Computational Physics, 1997, vol. 131, pp. 267-279. DOI: 10.1006/jcph.1996.5572
6. Arnold D.N., Brezzi F., Cockburn B., Marini L.D. Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems. SIAM Journal on Numerical Analysis, 2002, vol. 29, pp. 1749-1779. DOI: 10.1137/S0036142901384162
7. Cockburn B., Shu C.W. Runge - Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems. Journal of Scientific Computing, 2001, vol. 3, pp. 173-261. DOI: 10.1023/A:1012873910884
8. Ladonkina M.E., Tishkin V.F. On the Connection of Discontinuous Galerkin Method and Godunov Type Methods of High Order Accuracy. Preprints Keldysh Institute of Applied Mathematics, 2014, no. 49, 10 p. Available at: http://library.keldysh.ru/preprint.asp?id=2014-49 (in Russian)