Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-26T19:33:04.240Z Has data issue: false hasContentIssue false
  • English
  • Français

Error Analysis and Adaptive Methods of Least Squares Nonconforming Finite Element for the Transport Equations

Part of: Partial differential equations, boundary value problems

Published online by Cambridge University Press:  08 July 2016

Huipo Liu ,
Shuanghu Wang  and
Hongbin Han
Huipo Liu*
Affiliation:
Institute of Applied Physics and Computational Mathematics, Beijing 100094, China
Shuanghu Wang
Affiliation:
Institute of Applied Physics and Computational Mathematics, Beijing 100094, China
Hongbin Han
Affiliation:
Department of Radiology, Peking University Third Hospital, Beijing 100191, China Beijing Key Lab of Magnetic Resonance Imaging Technology, Beijing 100191, China
*
*Corresponding author. Email:[email protected] (H. P. Liu)
Get access

Abstract

In this paper, we consider a least squares nonconforming finite element of low order for solving the transport equations. We give a detailed overview on the stability and the convergence properties of our considered methods in the stability norm. Moreover, we derive residual type a posteriori error estimates for the least squares nonconforming finite element methods under H–1-norm, which can be used as the error indicators to guide the mesh refinement procedure in the adaptive finite element method. The theoretical results are supported by a series of numerical experiments.

Keywords

Least squaresnonconformingtransport equationsadaptive methods

MSC classification

Secondary: 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 8 , Issue 5 , October 2016 , pp. 871 - 886
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Adams, R. A., Sobolev Spaces, Academic Press, New York, 1975.Google Scholar
[2]Asadzadeh, M., Analysis of a fully discrete scheme for neutron transport in two-dimensional geometry, SIAM J. Numer. Anal., 23(1986), pp. 543–561.CrossRefGoogle Scholar
[3]Baccouch, M. and Adjerid, S., Discontinuous Galerkin error estimation for hyperbolic problems on unstructured triangular meshes, Comput. Methods Appl. Mech. Eng., 200 (2011), pp. 162–177.Google Scholar
[4]Bramble, H. and Scatz, A. H., Higher order local accuracy by averaging in the finite element method, Math. Comput., 31 (1977), pp. 94–111.Google Scholar
[5]Brenner, S. C. and Scott, L. R., The Mathematical Theory of Finite Element Methods, Springer, Berlin, 2002.Google Scholar
[6]Brezzi, F., Marini, D. and Süli, E., Residual-free bubbles for advection-diffusion problems: the general error analysis, Numer. Math., 85 (2000), pp. 31–47.Google Scholar
[7]Carstensen, C., Hu, J. and Orlando, A., Framework for the a posteriori error analysis of nonconforming finite elements, SIAM J. Numer. Anal., 45(2007), pp. 68–82.Google Scholar
[8]Ciarlet, P., The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.Google Scholar
[9]Clément, P., Approximation by finite element functions using local regularization, RAIRO Anal. Numér., 9 (1975), pp. 77–84.Google Scholar
[10]Cockburn, B., Dong, B., Guzmán, J. and Qian, J., Optimal convergence of the origial DG method on special meshes for variable transport velocity, SIAM J. Nmer. Anal., 48 (2010), pp. 133–146.CrossRefGoogle Scholar
[11]Courbet, B. and Croisille, J., Finite volume box schemes on triangular meshes, ESAIM Math. Mod. Numer. Anal., 32 (1998), pp. 631–649.Google Scholar
[12]Croisille, J., Finite volume box schemes and mixed methods, ESAIM Math. Mod. Numer. Anal., 31 (2000), pp. 1087–1106.Google Scholar
[13]Crouzeix, M. and Raviart, P. A., Conforming and nonconforming finite element methods for solving the stationary Stokes equations, RAIRO Anal. Numer., 7 (1973), pp. 33–76.Google Scholar
[14]El Alaoui, L. and Ern, A., Nonconforming finite element methods with subgrid viscosity applied to advection-diffusion-reaction equations, Numer. Methods Partial Differential Equations, 22 (2006), pp. 1106–1126.Google Scholar
[15]Houston, P., Rannacher, R. and Süli, E., A posteriori error analysis for stabilised finite element approximations of transport problems, Comput. Methods Appl. Mech. Eng., 190 (2000), pp. 1483–1508.Google Scholar
[16]Houston, P., Mackenzie, J., Süli, E. and Warnecke, G., A posteriori error analysis for numerical approximations of Friedrichs systems, Numer. Math., 82 (1999), pp. 409–432.Google Scholar
[17]John, V., Maubach, J. M. and Tobiska, L., Nonconforming streamline-diffusion-finite-element-methods for convection-diffusion problems, Numer. Math., 78 (1997), pp. 165–188.Google Scholar
[18]John, V., Matthies, G., Schieweck, F. and Tobiska, L., A streamline-diffusion method for nonconforming finite element approximations applied to convection-diffusion problems, Comput. Methods Appl. Mech. Eng., 166 (1998), pp. 85–97.CrossRefGoogle Scholar
[19]Johnson, C., Nävert, U. and Pitkäranta, J., Finite element methods for linear hyperbolic problems, Comput. Methods Appl. Mech. Eng., 45 (1984), pp. 285–312.CrossRefGoogle Scholar
[20]Matthies, G. and Tobiska, L., The stream-diffusion method for conforming and nonconforming finite elements of lowest order applied to convection-diffusion problems, Computing, 66 (2001), pp. 343–364.CrossRefGoogle Scholar
[21]Rauch, J., L2 is a continuable initial condition for Kress' mixed problems, Commun. Pure Appl. Math., 25 (1972), pp. 265–285.CrossRefGoogle Scholar
[22]Reed, W. H. and Hill, T. R., Triangular mesh methods for the neutron transport equation, Tech. Report LA-UR-73-479, Los Alamos Scientific Laboratory, 1973.Google Scholar
[23]Scott, L. R. and Zhang, S., Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comput., 54 (1990), pp. 483–493.CrossRefGoogle Scholar
[24]Süli, E., The accuracy of cell vertex finite volume methods on quadrilateral meshes, Math. Comput., 59 (1992), pp. 359–382.Google Scholar
[25]Zhang, Z., Analysis of some quadrilateral nonconforming elements for incompressible elasticity, SIAM J. Numer. Anal., 34 (1997), pp. 640–663.Google Scholar
[26]Zhang, T. and Feng, N., A posteriori error analysis of the discontinuous finite element methods for first order hyperbolic problems, Appl. Math. Comput., 218 (2011), pp. 1752–1764.Google Scholar