hp-version a priori error analysis of interior penalty discontinuous Galerkin finite element approximations to the biharmonic equation

Journal of Scientific Computing

Volumn 30, Issue 3, 2007, Pages 465-491

  Mozolevski, Igor ^a   Süli, Endre ^b   Bösing, Paulo R ^c  

^a FEDERAL UNIVERSITY OF SANTA CATARINA   (Brazil)

^b UNIVERSITY OF OXFORD   (United Kingdom)

^c UNIVERSITY OF SÃO PAULO   (Brazil)

Author keywords

A priori error analysis; Discontinuous Galerkin methods; Finite element methods; High order elliptic equations; Linear functionals

Indexed keywords

A PRIORI ERROR ANALYSIS; BIHARMONIC EQUATION; DISCONTINUOUS GALERKIN METHODS; HIGH-ORDER ELLIPTIC EQUATIONS; LINEAR FUNCTIONALS;

APPROXIMATION THEORY; COMPUTATIONAL COMPLEXITY; FINITE ELEMENT METHOD; GALERKIN METHODS; POISSON EQUATION; POLYNOMIALS;

ERROR ANALYSIS;

EID: 33947419735     PISSN: 08857474     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10915-006-9100-1     Document Type: Article

References (27)

1. Arnold, D. N., Brezzi, F., Cockburn, B., and Marini, D. (2001). Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749-1779.
2. Babuška, I., and Miller, A. (1984). The post-processing approach in the finite element method - part 1: Calculation of displacements, stresses and other higher derivatives of the displacements. Int. J. Numer. Meth. Eng. 20(6), 1085-1109.
3. Babuška, I., and Miller, A. (1984). The post-processing approach in the finite element method - part 2: The calculation of stress intensity factors. Int. J. Numer. Meth. Eng. 20(6), 1111-1129.
4. Babuška, I., and Miller, A. (1984). The post-processing approach in the finite element method - part 3: A posteriori error estimates and adaptive mesh selection. Int. J. Num. Meth. Eng. 20(6), 2311-2325.
5. Babuška, I., and Suri, M. (1987). The hp-version of the finite element method with quasiuniform meshes. Math. Model. Numer. Anal. 21, 199-238.
6. Baker, G. (1977). Finite element methods for elliptic equations using nonconforming elements. Math. Camp. 31, 44-59.
7. Blum, H., and Rannacher, R. (1980). On the boundary value problem of the biharmonic operator on domains with angular corners. Math. Meth. Appl. Sci. 2, 556-581.
8. Brenner, S. C., and Scott, L. R. (2002). The Mathematical Theory of Finite Element Methods, 2nd edn. Spinger, Berlin.
9. Brezzi, F., and Fortin, M. (1991). Mixed and Hybrid Finite Element Methods. Springer, Berlin.
10. Ciarlet, P. G. (1991). Basic error estimates for elliptic problems. In Ciarlet, P., and Lions, J. (eds.), Handbook of Numerical Analysis, Vol. 2. North Holland, Amsterdam.
11. Cockburn, B., Karniadakis, G. E., and Shu, C.-W (eds.) (2000). Discontinuous Galerkin Methods, LNCSE, Vol. 11, Springer, Berlin.
12. Douglas, J. J., and Dupont, T. (1976). Interior penalty procedures for elliptic and parabolic Galerkin methods. Lectures Notes in Physics, Vol. 58. Springer-Verlag, Berlin.
13. Engel, G., Garikipati, K., Hughes, T., Larson, M., Mazzei, L., and Taylor, R. (2002). Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity. Comput. Meth. Appl. Mech. Eng. 191, 3669-3750.
14. Georgoulis, E. H., and Süli, E. (2005). Optimal error estimates for the hp-version interior penalty discontinuous Galerkin finite element method. IMA J. Numer. Anal. 25, 205-220.
15. Girault, V., and Raviart, P. (1986). Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms. Spring-Verlag, Berlin.
16. Grisvard, P. (1992). Singularities in Boundary Value Problems. Research Notes in Applied Mathematics. Masson, Paris.
17. Harriman, K., Gavaghan, D. J., and Süli, E. (2004) The Importance of Adjoint Consistency in the Approximations of Linear Functionals Using the Discontinuous Galerkin Finite Element Method. Tech. Rep. 04/18, Oxford University Computing Laboratory.
18. Harriman, K., Houston, P., Senior, B., and Süli, E. (2002). hp-version discontinuous Galerkin methods with interior penalty for partial differential equations with nonnegative characteristic form. In Shu, C.-W., Tang, T., and Cheng, S.-Y. (eds.), Recent Advances in Scientific Computing and Partial Differential Equations, Contemporary Mathematics Vol. 330, AMS, 2003. pp. 89-119.
19. Kanschat, G., and Rannacher, R. (2002). Local error analysis of the interior penalty discontinuous Galerkin method for second order elliptic problems. J. Numer. Math. 10(4), 249-274.
20. Kozlov, V. A., and Maz'ya, V. G. (1996). Singularities in solutions to mathematical physics problems in non-smooth domains. In Cea, J. et al. (ed.), Partial Differential Equations and Functional Analysis. Proceedings of the conference held in November 1994 in memory of Pierre Grisvard. Progr. Nonlinear Differential Equations Appl., Vol. 22, Birkhäuser Boston, Boston, MA. pp. 174-206.
21. Larson, M., and Niklasson, A. (2004). Analysis of a family of discontinuous Galerkin methods for elliptic problems: the one dimensional case. Numer. Math. 99(1), 113-130.
22. Mozolevski, I., and Süli, E. (2003). A priori error analysis for the hp-version of the discontinuous Galerkin finite element method for the biharmonic equation. Comput. Meth. Appl. Math. 3(4), 596-607.
23. Oden, J. T., Babuška, I., and Baumann, C. (1998). The local discontinuous Galerkin finite element method for diffusion problems. J. Comput. Phys. 146, 491-519.
24. Rivière, B., Wheeler, M. F., and Girault, V. (2001). A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems. J. Numer. Anal. 39(3), 902-931.
25. Schwab, C. (1998). p- and hp- Finite Element Methods. Theory and Applications to Solid and Fluid Mechanics. Oxford University Press, Oxford.
26. Süli, E., and Mozolevski, I. (2004). hp-version Interior Penalty DGFEMs for the Biharmonic Equation. Techinical Report 04/05, Oxford University Computing Laboratory. http://web.comlab.ox.ac.uk/oucl/publications/ natr/na-04-05.html
27. Timoshenko, S. P., and Woinowsky-Krieger, S. (1984). Theory of plates and shells, 25 edn. McGraw-Hill International Book Company, New york.