Continuous interior penalty ℎ𝑝-finite element methods for advection and advection-diffusion equations

Burman, Erik; Ern, Alexandre

doi:10.1090/S0025-5718-07-01951-5

Continuous interior penalty -finite element methods for advection and advection-diffusion equations

Authors: Erik Burman and Alexandre Ern
Journal: Math. Comp. 76 (2007), 1119-1140
MSC (2000): Primary 65N30, 65N12, 65N15, 65D05, 65N35
DOI: https://doi.org/10.1090/S0025-5718-07-01951-5
Published electronically: January 24, 2007
MathSciNet review: 2299768
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A continuous interior penalty -finite element method that penalizes the jump of the gradient of the discrete solution across mesh interfaces is introduced. Error estimates are obtained for advection and advection-diffusion equations. The analysis relies on three technical results that are of independent interest: an -inverse trace inequality, a local discontinuous to continuous -interpolation result, and -error estimates for continuous -orthogonal projections.

1. Ivo Babuška and Milo R. Dorr, Error estimates for the combined ℎ and 𝑝 versions of the finite element method, Numer. Math. 37 (1981), no. 2, 257–277. MR 623044, https://doi.org/10.1007/BF01398256
2. I. Babuška and Manil Suri, The ℎ-𝑝 version of the finite element method with quasi-uniform meshes, RAIRO Modél. Math. Anal. Numér. 21 (1987), no. 2, 199–238 (English, with French summary). MR 896241, https://doi.org/10.1051/m2an/1987210201991
3. Ivo Babuška and Miloš Zlámal, Nonconforming elements in the finite element method with penalty, SIAM J. Numer. Anal. 10 (1973), 863–875. MR 345432, https://doi.org/10.1137/0710071
4. Garth A. Baker, Finite element methods for elliptic equations using nonconforming elements, Math. Comp. 31 (1977), no. 137, 45–59. MR 431742, https://doi.org/10.1090/S0025-5718-1977-0431742-5
5. Erik Burman, A unified analysis for conforming and nonconforming stabilized finite element methods using interior penalty, SIAM J. Numer. Anal. 43 (2005), no. 5, 2012–2033. MR 2192329, https://doi.org/10.1137/S0036142903437374
6. Erik Burman and Alexandre Ern, Stabilized Galerkin approximation of convection-diffusion-reaction equations: discrete maximum principle and convergence, Math. Comp. 74 (2005), no. 252, 1637–1652. MR 2164090, https://doi.org/10.1090/S0025-5718-05-01761-8
7. -, Continuous interior penalty -finite element methods for transport operators, Numerical Mathematics and Advanced Applications (Berlin), Springer, 2006, Enumath 2005 Conf. Proc.
8. Erik Burman, Miguel A. Fernández, and Peter Hansbo, Continuous interior penalty finite element method for Oseen’s equations, SIAM J. Numer. Anal. 44 (2006), no. 3, 1248–1274. MR 2231863, https://doi.org/10.1137/040617686
9. Erik Burman and Peter Hansbo, Edge stabilization for Galerkin approximations of convection-diffusion-reaction problems, Comput. Methods Appl. Mech. Engrg. 193 (2004), no. 15-16, 1437–1453. MR 2068903, https://doi.org/10.1016/j.cma.2003.12.032
10. C. Canuto and A. Quarteroni, Approximation results for orthogonal polynomials in Sobolev spaces, Math. Comp. 38 (1982), no. 157, 67–86. MR 637287, https://doi.org/10.1090/S0025-5718-1982-0637287-3
11. Philip J. Davis and Philip Rabinowitz, Methods of numerical integration, 2nd ed., Computer Science and Applied Mathematics, Academic Press, Inc., Orlando, FL, 1984. MR 760629
12. Jim Douglas Jr. and Todd Dupont, Interior penalty procedures for elliptic and parabolic Galerkin methods, Computing methods in applied sciences (Second Internat. Sympos., Versailles, 1975) Springer, Berlin, 1976, pp. 207–216. Lecture Notes in Phys., Vol. 58. MR 0440955
13. Linda El Alaoui and Alexandre Ern, Residual and hierarchical a posteriori error estimates for nonconforming mixed finite element methods, M2AN Math. Model. Numer. Anal. 38 (2004), no. 6, 903–929. MR 2108938, https://doi.org/10.1051/m2an:2004044
14. A. Ern and J.-L. Guermond, Discontinuous Galerkin methods for Friedrichs’ systems. I. General theory, SIAM J. Numer. Anal. 44 (2006), no. 2, 753–778. MR 2218968, https://doi.org/10.1137/050624133
15. K. O. Friedrichs, Symmetric positive linear differential equations, Comm. Pure Appl. Math. 11 (1958), 333–418. MR 100718, https://doi.org/10.1002/cpa.3160110306
16. Ronald H. W. Hoppe and Barbara Wohlmuth, Element-oriented and edge-oriented local error estimators for nonconforming finite element methods, RAIRO Modél. Math. Anal. Numér. 30 (1996), no. 2, 237–263 (English, with English and French summaries). MR 1382112, https://doi.org/10.1051/m2an/1996300202371
17. P. Houston, D. Schötzau, and Th.P. Wihler, Energy norm a posteriori error estimation of -adaptive Discontinuous Galerkin methods for elliptic problems, Tech. Report IMA Preprint Series 1985, Institute for Mathematics and its Applications, 2004.
18. Paul Houston, Christoph Schwab, and Endre Süli, Stabilized ℎ𝑝-finite element methods for first-order hyperbolic problems, SIAM J. Numer. Anal. 37 (2000), no. 5, 1618–1643. MR 1759909, https://doi.org/10.1137/S0036142998348777
19. Paul Houston, Christoph Schwab, and Endre Süli, Discontinuous ℎ𝑝-finite element methods for advection-diffusion-reaction problems, SIAM J. Numer. Anal. 39 (2002), no. 6, 2133–2163. MR 1897953, https://doi.org/10.1137/S0036142900374111
20. P. Houston and E. Süli, Stabilised ℎ𝑝-finite element approximation of partial differential equations with nonnegative characteristic form, Computing 66 (2001), no. 2, 99–119. Archives for scientific computing. Numerical methods for transport-dominated and related problems (Magdeburg, 1999). MR 1825801, https://doi.org/10.1007/s006070170030
21. M. Jensen, Discontinuous Galerkin methods for Friedrichs systems with irregular solutions, Ph.D. thesis, University of Oxford, Oxford, UK, 2004.
22. Ohannes A. Karakashian and Frederic Pascal, A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems, SIAM J. Numer. Anal. 41 (2003), no. 6, 2374–2399. MR 2034620, https://doi.org/10.1137/S0036142902405217
23. J. M. Melenk, ℎ𝑝-interpolation of nonsmooth functions and an application to ℎ𝑝-a posteriori error estimation, SIAM J. Numer. Anal. 43 (2005), no. 1, 127–155. MR 2177138, https://doi.org/10.1137/S0036142903432930
24. Alfio Quarteroni and Alberto Valli, Numerical approximation of partial differential equations, Springer Series in Computational Mathematics, vol. 23, Springer-Verlag, Berlin, 1994. MR 1299729
25. Ch. Schwab, 𝑝- and ℎ𝑝-finite element methods, Numerical Mathematics and Scientific Computation, The Clarendon Press, Oxford University Press, New York, 1998. Theory and applications in solid and fluid mechanics. MR 1695813
26. T. Warburton and J. S. Hesthaven, On the constants in ℎ𝑝-finite element trace inverse inequalities, Comput. Methods Appl. Mech. Engrg. 192 (2003), no. 25, 2765–2773. MR 1986022, https://doi.org/10.1016/S0045-7825(03)00294-9