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Article information
2016 , Volume 21, ¹ 4, p.49-63
Itkina N.B., Markov S.I.
Discontinuous Galerkin Method for solution of singularly perturbed problems
The objective of our paper is to develop and verify a stable computational finiteelement scheme for solution of singularly perturbed problems. In this paper we present a nonconformal finite element method (the Discontinuous Galerkin Method) using special lifting-operators that enhance the stability of the computational scheme. The computational scheme with special stabilizing terms and lifting-operators is proposed. The verification of the computational scheme is carried out in the model problem class. The optimal change range of the Bassi and Arnold stabilizers was defined. The efficacy of the ℎ-refinement technology application for solving the singularly perturbed problems was substantiated. A high flexibility Discontinuous Galerkin Method allows you to apply this computational scheme for solving the problems with singular perturbations and boundary layers. It was found that the optimal choice of stabilization parameter can effectively apply the ℎ-refinement technology to find solutions and significantly reduce the dimension of the matrix linear systems. Because of the local conservatism, all the computing circuits based on the Discontinuous Galerkin Method give an adequate idea for solving the problem on a fairly coarse tessellation computational domain. A sharp increase in the dimension of the SLAE matrix using the high-order basis functions and a high computational scheme sensitivity to the choice the stabilizing parameters is the main disadvantages of the computational scheme.
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Keywords: The Discontinuous Galerkin Method, singularly perturbed problem purpose
Author(s):
Itkina Natalya Borisovna
PhD. , Associate Professor
Position: Associate Professor
Office: Novosibirsk State Technical University
Address: 630073, Russia, Novosibirsk
Phone Office: (383) 346-27-76
E-mail: itkina.nat@yandex.ru
Markov Sergey Igorevich
Position: Student
Office: Novosibirsk State Technical University
Address: 630073, Russia, Novosibirsk
E-mail: www.sun91@list.ru
References:
[1] Cocburn, B. Superconvergence of the numerical traces of discontinuous Galerkin and hybridized mixed methods for convection-diffusion problems in one space dimension. Math. Comput. 2007; (76):67–96.
.
[2] Brenner, S.C., Scott, L.R. The mathematical theory of finite element methods.Texts in Applied Mathematics. Volume 15: 3rd ed. Springer; 2007: 420.
[3] Solin, P., Segeth, K., Dolezel, I. High-order finite element methods. Hardcover: Chapman & Hall/CRC Press; 2003: 408.
[4] Arnold, D.N., Brezzi, F., Cocburn, B., Marini D. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Analusis. 2002; 39(5):1749–1779.
[5] Baumann, C.E., Oden, J.T. A discontinuous hp finite element method for convectiondiffusion problems. Comput. Methods Appl. Mech. Eng. 2000; (175):311–341.
[6] Oden, J.T., Babuska, I., Baumann, C.E. A discontinuous hp finite element method for diffusion problems. J. of Comput. Physics. 1998; (146):491–519.
[7] Brezzi, F., Cockburn, B., Marini, L.D., Suli, E. Stabilization mechanisms in Discontinuous Galerkin finite element methods. Available at: http://eprints.maths.ox.ac.uk/1162/1/NA-04-24.pdf.
[8] Sudirham, J.J., van der Vegt, J.J.W., van Damme, R.M.J. A study on discontinuous Galerkin finite element methods for elliptic problems. Memorandum 1690. Mathematics of Computational Science (MaCS); 2003: 20. AvaLable at: http://purl.utwente.nl/publications/65875
[9] Gilbarg, D., Trudinger, N.S. Elliptic Partial Differential Equations of Second Order. Berlin; New York: Springer; 2001: 500.
[10] Sangalii, G. Capturing small scales in elliptic problems using a residual free bubbles finite element method. Multiscale Model Simul. 2003; (1):485–503.
Bibliography link:
Itkina N.B., Markov S.I. Discontinuous Galerkin Method for solution of singularly perturbed problems // Computational technologies. 2016. V. 21. ¹ 4. P. 49-63
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