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Journal of Computational Mathematics 2017, Vol. 35 Issue (5) :547-568    DOI: 10.4208/jcm.1605-m2015-0479
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Piotr Skrzypacz, Dongming Wei
Nazarbayev University, School of Science and Technology, 53 Kabanbay Batyr Ave., Astana 010000 Kazakhstan

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Abstract It is a well known fact that finite element solutions of convection dominated problems can exhibit spurious oscillations in the vicinity of boundary layers. One way to overcome this numerical instability is to use schemes that satisfy the discrete maximum principle. There are monotone methods for piecewise linear elements on simplices based on the upwind techniques or artificial diffusion. In order to satisfy the discrete maximum principle for the local projection scheme, we add an edge oriented shock capturing term to the bilinear form. The analysis of the proposed stabilisation method is complemented with numerical examples in 2D.
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KeywordsLocal projection stabilization   Discrete maximum principle   Shock capturing     
Received: 2015-12-18;
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[1] L. El Alaoui, A. Ern, and E. Burman. A nonconforming finite element method with face penalty for advection-diffusion equations. In Numerical mathematics and advanced applications, pages 512-519. Springer, Berlin, 2006.
[2] S. Badia and A. Hierro. On monotonicity-preserving stabilized finite element approximations of transport problems. SIAM J. Sci. Comput., 36(6):A2673-A2697, 2014.
[3] G. R. Barrenechea, E. Burman, and F. Karakatsani. Edge-based nonlinear diffusion for finite element approximations of convection-diffusion equations and its relation to algebraic flux-correction schemes. arXiv:1509.08636, 2015.
[4] J. H. Brandts, S. Korotov, and M. Krírek. The discrete maximum principle for linear simplicial finite element approximations of a reaction-diffusion problem. Linear Algebra Appl., 429:10(2008), 2344-2357.
[5] E. Burman. A unified analysis for conforming and nonconforming stabilized finite element methods using interior penalty. SIAM J. Numer. Anal., 43:5(2005), 2012-2033(electronic).
[6] E. Burman and A. Ern. Nonlinear diffusion and discrete maximum principle for stabilized Galerkin approximations of the convection-diffusion-reaction equation. Comput. Methods Appl. Mech. Engrg., 191:35(2002), 3833-3855.
[7] E. Burman and A. Ern. Stabilized Galerkin approximation of convection-diffusion-reaction equations:discrete maximum principle and convergence. Math. Comp., 74:252(2005), 1637-1652(electronic).
[8] P. G. Ciarlet and P.-A. Raviart. Maximum principle and uniform convergence for the finite element method. Comput. Methods Appl. Mech. Engrg., 2(1973), 17-31.
[9] R. Codina. A discontinuity-capturing crosswind-dissipation for the finite element solution of the convection-diffusion equation. Comput. Methods Appl. Mech. Engrg., 110:3-4(1993), 325-342.
[10] A. Draganescu, T. F. Dupont, and L. Ridgway Scott. Failure of the discrete maximum principle for an elliptic finite element problem. Math. Comp., 74:249(2005), 1-23(electronic).
[11] A. Ern and J.-L. Guermond. Theory and practice of finite elements, volume 159 of Applied Mathematical Sciences. Springer-Verlag, New York, 2004.
[12] L. C. Evans. Partial differential equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1998.
[13] L.P. Franca. An overview of the residual-free-bubbles method. In Numerical methods in mechanics (Concepción, 1995), volume 371 of Pitman Res. Notes Math. Ser., pages 83-92. Longman, Harlow, 1997.
[14] L.P. Franca and A. Russo. Deriving upwinding, mass lumping and selective reduced integration by residual-free bubbles. Appl. Math. Lett., 9:5(1996), 83-88.
[15] L.P. Franca and L. Tobiska. Stability of the residual free bubble method for bilinear finite elements on rectangular grids. IMA J. Numer. Anal., 22:1(2002), 73-87.
[16] P. Grisvard. Elliptic problems in nonsmooth domains, volume 24 of Monographs and Studies in Mathematics. Pitman (Advanced Publishing Program), Boston, MA, 1985.
[17] J.-L. Guermond. Stabilization of Galerkin approximations of transport equations by subgrid modeling. M2AN, 33:6(1999), 1293-1316.
[18] A. Hannukainen, S. Korotov, and T. Vejchodský. Discrete maximum principle for FE solutions of the diffusion-reaction problem on prismatic meshes. J. Comput. Appl. Math., 226:2(2009), 275-287.
[19] T.J.R. Hughes and A. Brooks. A multidimensional upwind scheme with no crosswind diffusion. In Finite element methods for convection dominated flows (Papers, Winter Ann. Meeting Amer. Soc. Mech. Engrs., New York, 34:AMD (1979), 19-35.
[20] T. Ikeda. Maximum principle in finite element models for convection-diffusion phenomena, volume 4 of Lecture Notes in Numerical and Applied Analysis. Kinokuniya Book Store Co. Ltd., Tokyo, 1983. North-Holland Mathematics Studies, 76.
[21] V. John and P. Knobloch. On spurious oscillations at layers diminishing (SOLD) methods for convection-diffusion equations. I. A review. Comput. Methods Appl. Mech. Engrg., 196:17-20(2007), 2197-2215.
[22] V. John and P. Knobloch. On spurious oscillations at layers diminishing (SOLD) methods for convection-diffusion equations. Ⅱ. Analysis for P1 and Q1 finite elements. Comput. Methods Appl. Mech. Engrg., 197:21-24(2008), 1997-2014.
[23] J. Karátson and S. Korotov. An algebraic discrete maximum principle in Hilbert space with applications to nonlinear cooperative elliptic systems. SIAM J. Numer. Anal., 47:4(2009), 2518-2549.
[24] J. Karátson, S. Korotov, and Michal Krírek. On discrete maximum principles for nonlinear elliptic problems. Math. Comput. Simulation, 76:1-3(2007), 99-108.
[25] P. Knobloch. Numerical solution of convection-diffusion equations using upwinding techniques satisfying the discrete maximum principle. In Proceedings of Czech-Japanese Seminar in Applied Mathematics 2005, volume 3 of COE Lect. Note, pages 69-76. Kyushu Univ. The 21 Century COE Program, Fukuoka, 2006.
[26] P. Knobloch and L. Tobiska. On the stability of finite-element discretizations of convectiondiffusion-reaction equations. IMA Journal of Numerical Analysis, pages 1-18(electronic), 2009. DOI:10.1093/imanum/drp020.
[27] D. Kuzmin, M. J. Shashkov, and D. Svyatskiy. A constrained finite element method satisfying the discrete maximum principle for anisotropic diffusion problems. J. Comput. Phys., 228:9(2009), 3448-3463.
[28] G. Matthies, P. Skrzypacz, and L. Tobiska. A unified convergence analysis for local projection stabilisations applied to the Oseen problem. M2AN Math. Model. Numer. Anal., 41:4(2007), 713-742.
[29] G. Matthies, P. Skrzypacz, and L. Tobiska. Stabilization of local projection type applied to convection-diffusion problems with mixed boundary conditions. Electron. Trans. Numer. Anal., 32(2008), 90-105.
[30] H.-G. Roos, M. Stynes, and L. Tobiska. Robust numerical methods for singularly perturbed differential equations, volume 24 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, second edition, 2008. Convection-diffusion-reaction and flow problems.
[31] P. Šolín and T. Vejchodský. A weak discrete maximum principle for hp-FEM. J. Comput. Appl. Math., 209:1(2007), 54-65.
[32] T. Vejchodský and P. Šolín. Discrete maximum principle for a 1D problem with piecewise-constant coefficients solved by hp-FEM. J. Numer. Math., 15:3(2007), 233-243.
[33] T. Vejchodský and P. Šolín. Discrete maximum principle for higher-order finite elements in 1D. Math. Comp., 76:260(2007), 1833-1846(electronic).
[34] T. Vejchodský and P. Šolín. Discrete maximum principle for Poisson equation with mixed bound-ary conditions solved by hp-FEM. Adv. Appl. Math. Mech., 1:2(2009), 201-214.
[35] J.-C. Xu and L. Zikatanov. A monotone finite element scheme for convection-diffusion equations. Math. Comp., 68:228(1999), 1429-1446.
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