Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-26T06:58:26.248Z Has data issue: false hasContentIssue false
  • English
  • Français

Coefficient Jump-Independent Approximation of the Conforming and Nonconforming Finite Element Solutions

Part of: Numerical approximation and computational geometry Partial differential equations, boundary value problems

Published online by Cambridge University Press:  08 July 2016

Shangyou Zhang
Shangyou Zhang*
Affiliation:
Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA
*
*Corresponding author. Email:[email protected] (S. Y. Zhang)
Get access

Abstract

A counterexample is constructed. It confirms that the error of conforming finite element solution is proportional to the coefficient jump, when solving interface elliptic equations. The Scott-Zhang operator is applied to a nonconforming finite element. It is shown that the nonconforming finite element provides the optimal order approximation in interpolation, in L2-projection, and in solving elliptic differential equation, independent of the coefficient jump in the elliptic differential equation. Numerical tests confirm the theoretical finding.

Keywords

Jump coefficientfinite elementL2 projectionweighted projectionScott-Zhang operator

MSC classification

Secondary: 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65D05: Interpolation
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 8 , Issue 5 , October 2016 , pp. 722 - 736
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]Arnold, D. N., Brezzi, F., Cockburn, B. and Marini, L. D., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39(5) (2001/02), pp. 1749–1779.Google Scholar
[2]Apel, T. and Dobrowolski, M., Anisotropic interpolation with applications to the finite element method, Computing, 47 (1992), pp. 277–293.Google Scholar
[3]Apel, T., Nicaise, S. and Schb¨erl, J., Crouzeix-Raviart type finite elements on anisotropic meshes, Numer. Math., 89(2) (2001), pp. 193–223.Google Scholar
[4]Beirao Da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L. D. and Russo, A., Basic principles of virtual element methods, Math. Models Methods Appl. Sci., 23(1) (2013), pp. 199–214.Google Scholar
[5]Bramble, J. H. and Xu, J., Some estimates for a weighted L2 projection, Math. Comput., 56 (1991), pp. 463–476.Google Scholar
[6]Brenner, S. C., Convergence of nonconforming multigrid methods, without full elliptic regularity, Math. Comput., 68 (1999), pp. 25–53.Google Scholar
[7]Brenner, S. C. and Scott, L. R., The Mathematical Theory of Finite Element Methods, Third Edition, Springer, New York, 2008.Google Scholar
[8]Brezzi, F., Douglas, J. Jr. and Marini, L. D., Two families of mixed finite elements for second order elliptic problems, Numer. Math., 47(2) (1985), pp. 217–235.Google Scholar
[9]Ciarlet, P. G., The Finite Element Method for Elliptic Problems, Studies in Mathematics and its Applications, Vol. 4. North-Holland Publishing Co., Amsterdam-NewYork-Oxford, 1978.CrossRefGoogle Scholar
[10]Cockburn, B. and Gopalakrishnan, J., Error analysis of variable degree mixed methods for elliptic problems via hybridization, Math. Comput. 74(252) (2005), pp. 1653–1677.Google Scholar
[11]Demkowicz, L. and Gopalakrishnan, J., Analysis of the DPG method for the Poisson equation, SIAM J. Numer. Anal., 49(5) (2011), pp. 1788–1809.Google Scholar
[12]De Dios, B. A., Holst, M., Zhu, Y., and Zikatanov, L., Multilevel preconditioners for discontinuous, Galerkin approximations of elliptic problems, with jump coefficients, Math. Comput., 83(287) (2014), pp. 1083–1120.Google Scholar
[13]Mu, L., Wang, J., Ye, X. and Zhang, S., A C0-weak Galerkin finite element method for the biharmonic equation, J. Sci. Comput., 59(2) (2014), pp. 473–495.Google Scholar
[14]Raviart, P. A. and Thomas, J. M., A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods, Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975, pp. 292–315, Lecture Notes in Math., Vol. 606, Springer, Berlin, 1977.Google Scholar
[15]Scott, L. R. and Zhang, S., Finite-element interpolation of non-smooth functions satisfying boundary conditions, Math. Comput., 54 (1990), pp. 483–493.Google Scholar
[16]Wang, J. and Ye, X., A weak Galerkin mixed finite element method for second order elliptic problems, Math. Comput., 83(289) (2014), pp. 2101–2126.Google Scholar
[17]Xu, J., Theory of Multilevel Methods, Ph.D. thesis, Cornell Univ., 1989, AM report 48, Department of Mathematics, Penn State Univesity, July 1989.Google Scholar
[18]Xu, J., Counterexamples concerning a weighted L2projection, Math. Comput., 57 (1991), pp. 563–568.Google Scholar
[19]Zhu, Y. and Xu, J., Uniform convergent multigrid methods for elliptic problems with strongly discontinuous coefficients, Math. Models Methods Appl. Sci., 18 (2008), pp. 77–105.Google Scholar
[20]Zhang, S., Successive subdivisions of tetrahedra and multigrid methods on tetrahedral meshes, Houston J. Math., 21 (1995), pp. 541–556.Google Scholar