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A Posteriori Error Estimates of a Weakly Over-Penalized Symmetric Interior Penalty Method for Elliptic Eigenvalue Problems

Part of: Partial differential equations, boundary value problems

Published online by Cambridge University Press:  10 November 2015

Yuping Zeng ,
Jinru Chen  and
Feng Wang
Yuping Zeng*
Affiliation:
School of Mathematics, Jiaying University, Meizhou 514015, China
Jinru Chen
Affiliation:
Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China
Feng Wang
Affiliation:
Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China
*
*Corresponding author. Email addresses:[email protected](Y. Zeng), [email protected] (J. Chen), [email protected](F. Wang)
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Abstract

A weakly over-penalized symmetric interior penalty method is applied to solve elliptic eigenvalue problems. We derive a posteriori error estimator of residual type, which proves to be both reliable and efficient in the energy norm. Some numerical tests are provided to confirm our theoretical analysis.

Keywords

Interior penalty methodweakly over-penalizationelliptic eigenvalue problemsa posteriori error estimate

MSC classification

Secondary: 65N15: Error bounds 65N25: Eigenvalue problems 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 5 , Issue 4 , November 2015 , pp. 327 - 341
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Ainsworth, M., A posteriori error estimation for discontinuous Galerkin finite element approximation, SIAM J. Numer. Anal. 45, 1777–1798 (2007).Google Scholar
[2]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, 1749–1779 (2002).Google Scholar
[3]Barker, A.T. and Brenner, S.C., A mixed finite element method for the Stokes equations based on a weakly over-penalized symmetric interior penalty approach, J. Sci. Comput. 58, 290–307 (2014).CrossRefGoogle Scholar
[4]Babuška, I. and Osborn, J., Eigenvalue Problems, in Handbook of Numerical Analysis, Vol. II, North-Holland (1991).Google Scholar
[5]Becker, R., Hansbo, P. and Larson, M., Energy norm a posteriori error estimation for discontinuous Galerkin methods, Comput. Methods Appl. Mech. Eng. 192, 723–733 (2003).CrossRefGoogle Scholar
[6]Bösing, P.R. and Carstensen, C., Discontinuous Galerkin with weakly over-penalized techniques for Reissner-Mindlin plates, J. Sci. Comput. 64, 401–424 (2015).Google Scholar
[7]Brenner, S.C., Poincaré-Friedrichs inequalities for piecewise H1 functions, SIAM J. Numer. Anal. 41, 306–324 (2003).Google Scholar
[8]Brenner, S.C., Gudi, T. and Sung, L.Y., A posteriori error control for a weakly over-penalized symmetric interior penalty method, J. Sci. Comput. 40, 37–50 (2009).Google Scholar
[9]Brenner, S.C., Gudi, T. and Sung, L.Y., An intrinsically parallel finite element method, J. Sci. Comput. 42, 118–121 (2010).Google Scholar
[10]Brenner, S.C., Gudi, T. and Sung, L.Y., A weakly over-penalized symmetric interior penalty method for the biharmonic problem, Electron. Trans. Numer. Anal. 37, 214–238 (2010).Google Scholar
[11]Brenner, S.C., Owens, L. and Sung, L.Y., A weakly over-penalized symmetric interior penalty method, Electron. Trans. Numer. Anal., 30 (2008), 107–127.Google Scholar
[12]Brenner, S. C., Owens, L. and Sung, L.Y., Higher order weakly over-penalized symmetric interior penalty methods, J. Comput. Appl. Math. 236, 2883–2894 (2012).Google Scholar
[13]Carstensen, C. and Gedicke, J., An oscillation-free adaptive FEM for symmetric eigenvalue problems, Numer. Math. 118, 401–427 (2011).Google Scholar
[14]Chen, L. and Zhang, C., AFEM@matlab: a Matlab package of adaptive finite element methods,Technical report, University of Maryland (2006).Google Scholar
[15]Clément, P., Approximation by finite element functions using local regularisation, RAIRO Anal. Numér. 2, 77–84 (1975).Google Scholar
[16]Dai, X., Xu, J. and Zhou, A.. Convergence and optimal complexity of adaptive finite element eigenvalue computations, Numer. Math. 110, 313–355 (2008).Google Scholar
[17]Dari, E.A., Durán, R.G. and Padra, C., A posteriori error estimates for non-conforming approximation of eigenvalue problems, Appl. Numer. Math. 62, 580–591 (2012).Google Scholar
[18]Dörfler, W., A convergent adaptive algorithm for Poisson's equation, SIAM J. Numer. Anal. 33, 1106–1124 (1996).Google Scholar
[19]Durán, R.G. and Padra, C., A posteriori error estimators for mixed approximations of eigenvalue problems, Math. Models Meth. Appl. Sci. 9, 1165–1178 (1999).Google Scholar
[20]Durán, R.G., Padra, C. and Rodríguez, R., A posteriori error estimates for the finite element approximation of eigenvalue problems, Math. Models Meth. Appl. Sci. 13, 1219–1229 (2003).Google Scholar
[21]Ern, A. and Proft, J., A posteriori discontinuous Galerkin error estimates for transient convection-diffusion equations, Appl. Math. Lett. 18, 833–841 (2005).Google Scholar
[22]Garau, E.M., Morin, P. and Zuppa, C., Convergence of adaptive finite element methods for eigenvalue problems, Math. Models Meth. Appl. Sci. 19, 721–747 (2009).Google Scholar
[23]Giani, S. and Graham, I.G., A convergent adaptive method for elliptic eigenvalue problems, SIAM J. Numer. Anal. 47, 1067–1091 (2009).CrossRefGoogle Scholar
[24]Giani, S. and Hall, E.J.C., An a posteriori error estimator for hp-adaptive discontinuous Galerkin methods for elliptic eigenvalue problems, Math. Models Meth. Appl. Sci. 22, 1250030 (2012).Google Scholar
[25]Heuveline, V. and Rannacher, R., A posteriori error control for finite element approximations of elliptic eigenvalue problems, Adv. Comput. Math. 15, 107–138 (2001).CrossRefGoogle Scholar
[26]Hoppe, R.H.W., Wu, H. and Zhang, Z., Adaptive finite element methods for the Laplace eigenvalue problem, J. Numer. Math. 18, 281–302 (2010).Google Scholar
[27]Houston, P., Schötzau, D. and Wihler, T.P., Energy norm a posteriori error estimation of hp-adaptive discontinuous Galerkin methods for elliptic problems, Math. Models Meth. Appl. Sci. 17, 33–62 (2007).Google Scholar
[28]Jia, S., Chen, H. and Xie, H., A posteriori error estimator for eigenvalue problems by mixed finite element method, Sci. China Math. 56, 887–900 (2013).Google Scholar
[29]Karakashian, O.A. and Pascal, F., A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems, SIAM J. Numer. Anal. 41, 2374–2399 (2003).Google Scholar
[30]Larson, M.G., A posteriori and a priori error analysis for finite element approximations of self-adjoint elliptic eigenvalue problems, SIAM J. Numer. Anal. 38, 608–625 (2000).Google Scholar
[31]Lin, Q., Luo, F. and Xie, H., A posterior error estimator and lower bound of a nonconforming finite element method, J. Comput. Appl. Math. 265, 243–254 (2014).Google Scholar
[32]Rivière, B. and Wheeler, M.F., A posteriori error estimates for a discontinuous Galerkin method applied to elliptic problems, Comput. Math. Appl. 46, 141–163 (2003).Google Scholar
[33]Romkes, A., Prudhomme, S. and Oden, J.T., A posteriori error estimation for a new stabilised discontinuous Galerkin method, Appl. Math. Lett. 16, 447–452 (2003).Google Scholar
[34]Schneider, R., Xu, Y. and Zhou, A., An analysis of discontinuous Galerkin methods for elliptic problems, Adv. Comput. Math. 25, 259–286 (2006).Google Scholar
[35]Scott, L.R. and Zhang, S., Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comput. 54, 483–493 (1994).Google Scholar
[36]Verfürth, R., A Review of A Posteriori Estimation and Adaptive Mesh-Refinement Techniques, Wiley-Teubner, New York/Stuttgart (1996).Google Scholar
[37]Wohlmuth, B.I. and Hoppe, R.H.W., A comparison of a posteriori error estimators for mixed finite element discretizations by Raviart-Thomas elements, Math. Comput. 68, 1347–1378 (1999).Google Scholar
[38]Yang, J. and Chen, Y., A unified a posteriori error analysis for discontinuous Galerkin approximations of reactive transport equations, J. Comput. Math. 24, 425–434 (2006).Google Scholar
[39]Yang, Y., Han, J., Bi, H. and Yu, Y., The lower/upper boundproperty of the Crouzeix-Raviart element eigenvalues on adaptive meshes, J. Sci. Comput. 62, 284–299 (2015).Google Scholar
[40]Yang, Y., Zhang, Z. and Lin, F., Eigenvalue approximation from below using non-conforming finite elements, Sci. China Math. 53, 137–150 (2010).Google Scholar
[41]Zeng, Y., Chen, J. and Wang, F., Error estimates of the weakly over-penalized symmetric interior penalty method for two variational inequalities, Comput. Math. Appl. 69, 760–770 (2015).Google Scholar
[42]Zeng, Y., Chen, J., Wang, F. and Meng, Y., A priori and a posteriori error estimates of a weakly over-penalized interior penalty methodfor non-self-adjoint and indefinite problems, J. Comput. Math. 32, 332–347 (2014).Google Scholar