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Space-time discontinuous Galerkin method for solving nonstationary convection-diffusion-reaction problems. (English) Zbl 1164.65469
Summary: The paper presents the theory of the discontinuous Galerkin finite element method for the space-time discretization of a linear nonstationary convection-diffusion-reaction initial-boundary value problem. The discontinuous Galerkin method is applied separately in space and time using, in general, different nonconforming space grids on different time levels and different polynomial degrees $$p$$ and $$q$$ in space and time discretization, respectively. In the space discretization, the nonsymmetric interior and boundary penalty approximation of diffusion terms is used. The paper is concerned with the proof of error estimates in “$$L^2(L^2)$$”- and “$$\sqrt { \varepsilon } L^2(H^1)$$”-norms, where $$\varepsilon \geq 0$$ is the diffusion coefficient. Using special interpolation theorems for the space as well as time discretization, we find that under some assumptions on the shape regularity of the meshes and a certain regularity of the exact solution the errors are of order $$O(h^p+\tau ^q)$$. The estimates hold true even in the hyperbolic case when $$\varepsilon = 0$$.

MSC:
 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 35K15 Initial value problems for second-order parabolic equations
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References:
 [1] D. N. Arnold: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19 (1982), 742–760. · Zbl 0482.65060 [2] D. N. Arnold, F. Brezzi, B. Cockburn, and D. Marini: Discontinuos Galerkin methods for elliptic problems. In: Discontinuous Galerkin methods. Theory, Computation and Applications. Lect. Notes Comput. Sci. Eng. 11 (B. Cockburn et al., eds.). Springer-Verlag, Berlin, 2000, pp. 89–101. · Zbl 0948.65127 [3] D. N. Arnold, F. Brezzi, B. Cockburn, and D. Marini: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2002), 1749–1779. · Zbl 1008.65080 [4] I. Babuška, C. E. Baumann, and J.T. Oden: A discontinuous hp finite element method for diffusion problems, 1-D analysis. Comput. Math. Appl. 37 (1999), 103–122. · Zbl 0940.65076 [5] F. Bassi, S. Rebay: A high-order accurate discontinuous-nite element method for the numerical solution of the compressible Navier-Stokes equations. J. Comput. Phys. 131 (1997), 267–279. · Zbl 0871.76040 [6] F. Bassi, S. Rebay: High-order accurate discontinuous finite element solution of the 2D Euler equations. J. Comput. Phys. 138 (1997), 251–285. · Zbl 0902.76056 [7] C. E. Baumann, J.T. Oden: A discontinuous hp finite element method for the Euler and Navier-Stokes equations. Int. J. Numer. Methods Fluids 31 (1999), 79–95. · Zbl 0985.76048 [8] P. G. Ciarlet: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, 1978. [9] B. Cockburn: Discontinuous Galerkin methods for convection-dominated problems. In: High-Order Methods for Computational Physics. Lect. Notes Comput. Sci. Eng. 9 (T. J. Barth, H. Deconinck, eds.). Springer-Verlag, Berlin, 1999, pp. 69–224. · Zbl 0937.76049 [10] Discontinuous Galerkin Methods. Lect. Notes Comput. Sci. Eng. 11 (B. Cockburn, G. E. Karniadakis, and C.-W. Shu, eds.). Springer-Verlag, Berlin, 2000. · Zbl 0989.76045 [11] B. Cockburn, C.W. Shu: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II. General framework. Math. Comp. 52 (1989), 411–435. · Zbl 0662.65083 [12] T. A. Davis, I. S. Duff: A combined unifrontal/multifrontal method for unsymmetric sparse matrices. ACM Trans. Math. Softw. 25 (1999), 1–20. · Zbl 0962.65027 [13] V. Dolejší, M. Feistauer: On the discontinuous Galerkin method for the numerical solution of compressible high-speed flow. In: Numerical Mathematics and Advanced Applications, ENUMATH 2001 (F. Brezzi, A. Buffa, S. Corsaro, and A. Murli, eds.). Springer-Verlag, Berlin, 2003, pp. 65–83. [14] V. Dolejší, M. Feistauer: Error estimates of the discontinuous Galerkin method for nonlinear nonstationary convection-diffusion problems. Numer. Funct. Anal. Optimization 26 (2005), 349–383. · Zbl 1078.65078 [15] V. Dolejší, M. Feistauer: A semi-implicit discontinuous Galerkin finite element method for the numerical solution of inviscid compressible flow. J. Comput. Phys. 198 (2004), 727–746. · Zbl 1116.76386 [16] V. Dolejší, M. Feistauer, and J. Hozman: Analysis of semi-implicit DGFEM for nonlinear convection-diffusion problems on nonconforming meshes. Preprint No. MATHknm-2005/1. Charles University Prague, School of Mathematics, 2005; Comput. Methods Appl. Mech. Eng. In press. (doi:10.1016/j.cma.2006.09.025). [17] V. Dolejší, M. Feistauer, and V. Kučera: On a semi-implicit discontinuous Galerkin FEM for the nonstationary compressible Euler equations. In: Hyperbolic Problems: Theory, Numerics and Applications. I. Proc. 10th International Conference Osaka, September 2004 (F. Asakura, H. Aiso, S. Kawashima, A. Matsumura, S. Nishibata, and K. Nishihara, eds.). Yokohama Publishers, Yokohama, 2006, pp. 391–398. [18] V. Dolejší, M. Feistauer, and C. Schwab: A finite volume discontinuous Galerkin scheme for nonlinear convection-diffusion problems. Calcolo 39 (2002), 1–40. · Zbl 1098.65095 [19] V. Dolejší, M. Feistauer, and C. Schwab: On some aspects of the discontinuous Galerkin finite element method for conservation laws. Math. Comput. Simul. 61 (2003), 333–346. · Zbl 1013.65108 [20] V. Dolejší, M. Feistauer, and V. Sobotíková: A discontinuous Galerkin method for nonlinear convection-diffusion problems. Comput. Methods Appl. Mech. Eng. 194 (2005), 2709–2733. · Zbl 1093.76034 [21] J. Douglas, T. Dupont: Interior penalty procedures for elliptic and parabolic Galerkin methods. In: Computing methods in applied sciences (Second Internat. Sympos., Versailles, 1975). Lect. Notes Phys., Vol. 58. Springer-Verlag, Berlin, 1976, pp. 207–216. [22] K. Eriksson, D. Estep, P. Hansbo, and C. Johnson: Computational Differential Equations. Cambridge University Press, Cambridge, 1996. · Zbl 0946.65049 [23] K. Eriksson, C. Johnson, and V. Thomée: Time discretization of parabolic problems by the discontinuous Galerkin method. RAIRO, Modélisation Math. Anal. Numér. 19 (1985), 611–643. · Zbl 0589.65070 [24] M. Feistauer, V. Kučera: Solution of compressible flow with all Mach numbers. In: European Conference on Computational Fluid Dynamics, ECCOMAS CFD 2006 (P. Wesseling, E. Onate, and J. Périaux, eds.). TU Delft, The Netherlands, 2006, published electronically. [25] M. Feistauer, K. Švadlenka: Discontinuous Galerkin method of lines for solving nonstationary singularly perturbed linear problems. J. Numer. Math. 12 (2004), 97–117. · Zbl 1059.65083 [26] R. Hartmann, P. Houston: Adaptive discontinuous Galerkin finite element methods for the compressible Euler equations. Technical Report 2001-42 (SFB 359), IWR Heidelberg. [27] F. Hecht, O. Pironneau, and A. Le Hyaric: www.freefem.org/ff++. [28] P. Houston, C. Schwab, and E. Süli: Discontinuous hp-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 39 (2002), 2133–2163. · Zbl 1015.65067 [29] J. Jaffre, C. Johnson, and A. Szepessy: Convergence of the discontinuous Galerkin finite element method for hyperbolic conservation laws. Math. Models Methods Appl. Sci. 5 (1995), 367–386. · Zbl 0834.65089 [30] C. Johnson, J. Pitkäranta: An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comp. 46 (1986), 1–26. [31] A. Kufner, O. John, and S. Fučík: Function Spaces. Academia, Praha, 1977. [32] P. Le Saint, P.-A. Raviart: On a finite element method for solving the neutron transport equation. In: Mathematical Aspects of Finite Elements in Partial Differential Equations (C. de Boor, ed.). Academic Press, 1974, pp. 89–145. [33] J.-L. Lions: Quelques méthodes de résolution des problémes aux limites non liné aires. Dunod, Paris, 1969. [34] www.netlib.org/minpack. University of Chicago, Operator of Argonne Laboratory (1999). [35] J.T. Oden, I. Babuška, and C. E. Baumann: A discontinuous hp finite element method for diffusion problems. J. Comput. Phys. 146 (1998), 491–519. · Zbl 0926.65109 [36] W. H. Reed, T. R. Hill: Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73-479. Los Alamos Scientific Laboratory, 1973. [37] K. Rektorys: The Method of Discretization in Time and Partial Differential Equations. Reidel, Dordrecht, 1982. · Zbl 0522.65059 [38] B. Rivière, M.F. Wheeler: A discontinuous Galerkin method applied to nonlinear parabolic equations. In: Discontinuous Galerkin methods. Theory, Computation and Applications. Lect. Notes in Comput. Sci. Eng. 11 (B. Cockburn et al., eds.). Springer-Verlag, Berlin, 2000, pp. 231–244. · Zbl 0946.65078 [39] B. Rivière, M. F. Wheeler: Non-conforming methods for transport with nonlinear reaction. Contemp. Math. 295 (2002), 421–432. · Zbl 1068.76053 [40] B. Rivière, M.F. Wheeler, and V. Girault: Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. I. Comput. Geosci. 3 (1999), 337–360. · Zbl 0951.65108 [41] B. Rivière, M. F. Wheeler, and V. Girault: A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems. SIAM J. Numer. Anal. 39 (2001), 902–931. · Zbl 1010.65045 [42] H.-G. Roos, M. Stynes, and L. Tobiska: Numerical Methods for Singularly Perturbed Differential Equations. Convection-Diffusion and Flow Problems. Springer-Verlag, Berlin, 1996. · Zbl 0844.65075 [43] D. Schtözau: hp-DGFEM for Parabolic Evolution Problems. Applications to Diffusion and Viscous Incompressible Fluid Flow. PhD. Dissertation ETH No. 13041. ETH, Zürich, 1999. [44] D. Schötzau, C. Schwab: An hp a priori error analysis of the discontinuous Galerkin time-stepping for initial value problems. Calcolo 37 (2000), 207–232. · Zbl 1012.65084 [45] S. Sun, M.F. Wheeler: L 2(H 1)-norm a posteriori error estimation for discontinuous Galerkin approximations of reactive transport problems. J. Sci. Comput. 22–23 (2005), 501–530. · Zbl 1066.76037 [46] S. Sun, M.F. Wheeler: Symmetric and nonsymmetric discontinuous Galerkin methods for reactive transport in porous media. SIAM J. Numer. Anal. 43 (2005), 195–219. · Zbl 1086.76043 [47] S. Sun, M. F. Wheeler: Discontinuous Galerkin methods for coupled flow and reactive transport problems. Appl. Numer. Math. 52 (2005), 273–298. · Zbl 1079.76584 [48] J. J.W. van der Vegt, H. van der Ven: Space-time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flow. I. General formulation. J. Comput. Phys. 182 (2002), 546–585. · Zbl 1057.76553 [49] T. Werder, K. Gerdes, D. Schötzau, and C. Schwab: hp-discontinuous Galerkin time stepping for parabolic problems. Comput. Methods Appl. Mech. Eng. 190 (2001), 6685–6708. · Zbl 0992.65103 [50] M.F. Wheeler: An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal. 15 (1978), 152–161. · Zbl 0384.65058
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