Space-time discontinuos Galerkin method for solving nonstationary convection-diffusion-reaction problems

Miloslav Feistauer; Jaroslav Hájek; Karel Švadlenka

Applications of Mathematics (2007)

  • Volume: 52, Issue: 3, page 197-233
  • ISSN: 0862-7940

Abstract

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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 “ ε L 2 ( H 1 ) ”-norms, where ε ≥ 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 + τ q ) . The estimates hold true even in the hyperbolic case when ε = 0 .

How to cite

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Feistauer, Miloslav, Hájek, Jaroslav, and Švadlenka, Karel. "Space-time discontinuos Galerkin method for solving nonstationary convection-diffusion-reaction problems." Applications of Mathematics 52.3 (2007): 197-233. <http://eudml.org/doc/33285>.

@article{Feistauer2007,
abstract = {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 $\varepsilon \ge 0$ 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$.},
author = {Feistauer, Miloslav, Hájek, Jaroslav, Švadlenka, Karel},
journal = {Applications of Mathematics},
keywords = {nonstationary convection-diffusion-reaction equation; space-time discontinuous Galerkin finite element discretization; nonsymmetric treatment of diffusion terms; error estimates; error estimates; discontinuous Galerkin finite element method; linear nonstationary convection-diffusion-reaction initial-boundary value problem; time discretization; space discretization},
language = {eng},
number = {3},
pages = {197-233},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Space-time discontinuos Galerkin method for solving nonstationary convection-diffusion-reaction problems},
url = {http://eudml.org/doc/33285},
volume = {52},
year = {2007},
}

TY - JOUR
AU - Feistauer, Miloslav
AU - Hájek, Jaroslav
AU - Švadlenka, Karel
TI - Space-time discontinuos Galerkin method for solving nonstationary convection-diffusion-reaction problems
JO - Applications of Mathematics
PY - 2007
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 52
IS - 3
SP - 197
EP - 233
AB - 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 $\varepsilon \ge 0$ 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$.
LA - eng
KW - nonstationary convection-diffusion-reaction equation; space-time discontinuous Galerkin finite element discretization; nonsymmetric treatment of diffusion terms; error estimates; error estimates; discontinuous Galerkin finite element method; linear nonstationary convection-diffusion-reaction initial-boundary value problem; time discretization; space discretization
UR - http://eudml.org/doc/33285
ER -

References

top
  1. 10.1137/0719052, SIAM J. Numer. Anal. 19 (1982), 742–760. (1982) Zbl0482.65060MR0664882DOI10.1137/0719052
  2. 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. (2000) MR1842165
  3. 10.1137/S0036142901384162, SIAM J. Numer. Anal. 39 (2002), 1749–1779. (2002) MR1885715DOI10.1137/S0036142901384162
  4. 10.1016/S0898-1221(99)00117-0, Comput. Math. Appl. 37 (1999), 103–122. (1999) MR1688050DOI10.1016/S0898-1221(99)00117-0
  5. 10.1006/jcph.1996.5572, J.  Comput. Phys. 131 (1997), 267–279. (1997) MR1433934DOI10.1006/jcph.1996.5572
  6. 10.1006/jcph.1997.5454, J.  Comput. Phys. 138 (1997), 251–285. (1997) MR1607481DOI10.1006/jcph.1997.5454
  7. 10.1002/(SICI)1097-0363(19990915)31:1<79::AID-FLD956>3.0.CO;2-C, Int. J.  Numer. Methods Fluids 31 (1999), 79–95. (1999) MR1714511DOI10.1002/(SICI)1097-0363(19990915)31:1<79::AID-FLD956>3.0.CO;2-C
  8. The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. (1978) Zbl0383.65058MR0520174
  9. 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. (1999) Zbl0937.76049MR1712278
  10. Discontinuous Galerkin Methods. Lect. Notes Comput. Sci. Eng.  11, B.  Cockburn, G. E.  Karniadakis, and C.-W.  Shu (eds.), Springer-Verlag, Berlin, 2000. (2000) 
  11. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II.  General framework, Math. Comp. 52 (1989), 411–435. (1989) MR0983311
  12. 10.1145/305658.287640, ACM Trans. Math. Softw. 25 (1999), 1–20. (1999) MR1697461DOI10.1145/305658.287640
  13. 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. (2003) MR2360708
  14. 10.1081/NFA-200067298, Numer. Funct. Anal. Optimization 26 (2005), 349–383. (2005) MR2153838DOI10.1081/NFA-200067298
  15. 10.1016/j.jcp.2004.01.023, J.  Comput. Phys. 198 (2004), 727–746. (2004) MR2062915DOI10.1016/j.jcp.2004.01.023
  16. Analysis of semi-implicit  DGFEM for nonlinear convection-diffusion problems on nonconforming meshes, Preprint No.  MATH-knm-2005/1, Charles University Prague, School of Mathematics, 2005. (2005) MR2325393
  17. 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. (2006) MR2667262
  18. 10.1007/s100920200000, Calcolo 39 (2002), 1–40. (2002) MR1901200DOI10.1007/s100920200000
  19. 10.1016/S0378-4754(02)00087-3, Math. Comput. Simul. 61 (2003), 333–346. (2003) MR1984135DOI10.1016/S0378-4754(02)00087-3
  20. 10.1016/j.cma.2004.07.017, Comput. Methods Appl. Mech. Eng. 194 (2005), 2709–2733. (2005) MR2136396DOI10.1016/j.cma.2004.07.017
  21. 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. (1976) MR0440955
  22. Computational Differential Equations, Cambridge University Press, Cambridge, 1996. (1996) MR1414897
  23. 10.1051/m2an/1985190406111, RAIRO, Modélisation Math. Anal. Numér. 19 (1985), 611–643. (1985) MR0826227DOI10.1051/m2an/1985190406111
  24. 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. (2006) 
  25. Discontinuous Galerkin method of lines for solving nonstationary singularly perturbed linear problems, J. Numer. Math. 12 (2004), 97–117. (2004) MR2062581
  26. Adaptive discontinuous Galerkin finite element methods for the compressible Euler equations, Technical Report  2001-42 (SFB  359), IWR Heidelberg. MR1947780
  27. www.freefem.org/ff++, . 
  28. 10.1137/S0036142900374111, SIAM  J.  Numer. Anal. 39 (2002), 2133–2163. (2002) MR1897953DOI10.1137/S0036142900374111
  29. 10.1142/S021820259500022X, Math. Models Methods Appl. Sci. 5 (1995), 367–386. (1995) MR1330139DOI10.1142/S021820259500022X
  30. 10.1090/S0025-5718-1986-0815828-4, Math. Comp. 46 (1986), 1–26. (1986) MR0815828DOI10.1090/S0025-5718-1986-0815828-4
  31. Function Spaces, Academia, Praha, 1977. (1977) MR0482102
  32. 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. (1974) MR0658142
  33. Quelques méthodes de résolution des problémes aux limites non linéaires, Dunod, Paris, 1969. (1969) Zbl0189.40603MR0259693
  34. www.netlib.org/minpack, University of Chicago, Operator of Argonne Laboratory (1999). (1999) 
  35. 10.1006/jcph.1998.6032, J.  Comput. Phys. 146 (1998), 491–519. (1998) MR1654911DOI10.1006/jcph.1998.6032
  36. Triangular mesh methods for the neutron transport equation, Technical Report LA-UR-73-479. Los Alamos Scientific Laboratory, 1973. 
  37. The Method of Discretization in Time and Partial Differential Equations, Reidel, Dordrecht, 1982. (1982) Zbl0522.65059MR0689712
  38. 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. (2000) MR1842177
  39. 10.1090/conm/295/05032, Contemp. Math. 295 (2002), 421–432. (2002) MR1911967DOI10.1090/conm/295/05032
  40. 10.1023/A:1011591328604, Comput. Geosci. 3 (1999), 337–360. (1999) MR1750076DOI10.1023/A:1011591328604
  41. 10.1137/S003614290037174X, SIAM J.  Numer. Anal. 39 (2001), 902–931. (2001) MR1860450DOI10.1137/S003614290037174X
  42. Numerical Methods for Singularly Perturbed Differential Equations. Convection-Diffusion and Flow Problems, Springer-Verlag, Berlin, 1996. (1996) MR1477665
  43. h p -DGFEM for Parabolic Evolution Problems, Applications to Diffusion and Viscous Incompressible Fluid Flow. PhD.  Dissertation ETH No.  13041, ETH, Zürich, 1999. (1999) 
  44. 10.1007/s100920070002, Calcolo 37 (2000), 207–232. (2000) MR1812787DOI10.1007/s100920070002
  45. 10.1007/s10915-004-4148-2, J.  Sci. Comput. 22–23 (2005), 501–530. (2005) MR2142207DOI10.1007/s10915-004-4148-2
  46. 10.1137/S003614290241708X, SIAM J.  Numer. Anal. 43 (2005), 195–219. (2005) MR2177141DOI10.1137/S003614290241708X
  47. 10.1016/j.apnum.2004.08.035, Appl. Numer. Math. 52 (2005), 273–298. (2005) MR2116915DOI10.1016/j.apnum.2004.08.035
  48. 10.1006/jcph.2002.7185, J.  Comput. Phys. 182 (2002), 546–585. (2002) MR1941852DOI10.1006/jcph.2002.7185
  49. 10.1016/S0045-7825(01)00258-4, Comput. Methods Appl. Mech. Eng. 190 (2001), 6685–6708. (2001) MR1863353DOI10.1016/S0045-7825(01)00258-4
  50. 10.1137/0715010, SIAM J.  Numer. Anal. 15 (1978), 152–161. (1978) Zbl0384.65058MR0471383DOI10.1137/0715010

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