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A Priori and a Posteriori Error Estimates for H(div)-Elliptic Problem with Interior Penalty Method

Published online by Cambridge University Press:  03 June 2015

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Abstract

In this paper, we propose and analyze the interior penalty discontinuous Galerkin method for H(div)-elliptic problem. An optimal a priori error estimate in the energy norm is proved. In addition, a residual-based a posteriori error estimator is obtained. The estimator is proved to be both reliable and efficient in the energy norm. Some numerical testes are presented to demonstrate the effectiveness of our method.

Type
Research Article
Copyright
Copyright © Global Science Press Limited 2013

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References

[1]Ainsworth, M., A posteriori error estimation for discontinuous Galerkin finite element approximation, SIAM J. Numer. Anal., 45(2007), 17771798.CrossRefGoogle Scholar
[2]Ainsworth, M. and Oden, J. T., A Posteriori Error Estimation in Finite Element Analysis, Wiley-Interscience, JohnWiley & Sons, New York, 2000.CrossRefGoogle Scholar
[3]Arnold, D. N., An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal., 19(1982), 742760.CrossRefGoogle Scholar
[4]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(2002), 17491779.CrossRefGoogle Scholar
[5]Arnold, D. N., Falk, R. S. and Winther, R., Multigrid preconditioning in H(div) and application, Math. Comp., 66(1997), 957984.CrossRefGoogle Scholar
[6]Babuska, I. and Strouboulis, T., The Finite Element Method and its Reliability, Clarendon Press, Oxford University Press, New York, 2001.Google Scholar
[7]Becker, R., Hansbo, P. and Larson, M., Energy norm a posteriori error estimation for discontinuous Galerkin methods, Comput. Methods Appl. Mech. Engrg., 192(2003), 723733.CrossRefGoogle Scholar
[8]Beck, R., Hiptmair, R., Hoppe, R. H. W. and Wohlmuth, B. I., Residual based a-posteriori error estimators for eddy current computation, M2AN. Math. Model. Numer. Anal., 34(2000), 159182.CrossRefGoogle Scholar
[9]Brenner, S. C. and Scott, L. R., The Mathematical Theory of Finite Element Methods (3nd edn), Springer-Verlag, New York/Berlin/Heidelberg, 2008.CrossRefGoogle Scholar
[10]Brezzi, F., Douglas, J. and Marini, L. D., Two families of mixed finite elements for second order elliptic problems, Numer. Math., 47(1985), 217235.CrossRefGoogle Scholar
[11]Brezzi, F. and Fortin, M., Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991.CrossRefGoogle Scholar
[12]Cai, Z., Lazarov, R., Manteuffel, T. and McCormick, S., First-order system least-squares for partial differential equations, Part I. SIAM J. Numer. Anal., 31(1994), 17851799.CrossRefGoogle Scholar
[13]Cascon, J. M., Nochetto, R. H. and Siebert, K. G., Design and convergence of AFM in H(div), Math. Models Methods Appl. Sci., 17(2007), 18491881.CrossRefGoogle Scholar
[14]Chen, L. and Zhang, C., A MATLAB Package of Adaptive Finite Element Methods, Technical report, University of Maryland, College Park, MD, 2006.Google Scholar
[15]Ciarlet, P., The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.Google Scholar
[16]Cleément, P., Approximation by finite element functions using local regularization,RAIRO Anal. Numér., 2(1975), 7784.Google Scholar
[17]Cockburn, B., Karniadakis, G.E. and Shu, C.W. (Eds.), Discontinuous Galerkin Methods-Theory, Computation and Applications, Lecture Notes in Computational Science and Engineering 11. Springer-Verlag, New York, 2000.Google Scholar
[18]Girault, V. and Raviart, P. A., Finite element methods for Navier-Stokes equations, Springer-Verlag, New York, 1986.CrossRefGoogle Scholar
[19]Hiptmair, R. and Xu, J., Nodal auxiliary space preconditioning in H(curl) and H(div) spaces. SIAM J. Numer. Anal., 45(2007), 24832509.CrossRefGoogle Scholar
[20]Houston, P., Perugia, I., Schneebeli, A. and Schoätzau, D., Interior penaltymethod for the indefinite time-harmonic Maxwell equations. Numer. Math., 100(2005), 485518.CrossRefGoogle Scholar
[21]Houston, P., Perugia, I. and Schoätzau, D., An a posteriori error indicator for discontinuous Galerkin discretizations of H(curl)-elliptic partial differential equations, IMA J. Numer. Anal., 27 (2007), 122150.CrossRefGoogle Scholar
[22]Houston, P., Schoätzau, D. and Wihler, T. P., Energy norm a posteriori error estimation of hp-adaptive discontinuous Galerkin methods for elliptic problems, Math. Models Methods Appl. Sci. 17 (2007), 3362.CrossRefGoogle Scholar
[23]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 (2003), 23742399.CrossRefGoogle Scholar
[24]Lin, P., A sequential regularization method for time-dependent incompressible Navier-Stokes equations, SIAM J. Numer. Anal., 34 (1997), 10511071.CrossRefGoogle Scholar
[25]Monk, P., Finite Element Methods for Maxwell’s Equation, Oxford University press, New York, 2003.CrossRefGoogle Scholar
[26]Neédélec, J. C., Mixed finite elements in R 3, Numer. Math., 35(1980), 315341.CrossRefGoogle Scholar
[27]Neédélec, J. C., A new family of mixed finite elements in R 3, Numer. Math., 50(1986), 5781.CrossRefGoogle Scholar
[28]Neittaanmaäki, P. and Repin, S., Reliable Methods for Mathematical Modelling. Error Control and a Posteriori Estimates, Elsevier, New York, 2004.Google Scholar
[29]Perugia, I. and Schoätzau, D., The hp-local discontinuous Galerkin method for low-frequency time-harmonic Maxwell equations. Math. Comp., 72(2003), 11791214.CrossRefGoogle Scholar
[30]Raviart, P. A. and Thomas, J. M., A mixed finite element method for second order elliptic problems, Lecture Notes in Math., Berlin-Heidelberg Newyork Springer, Vol. 66, 1977, 292315.Google Scholar
[31]Rivie, B.re, Wheeler, M. F. and Girault, V., A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems, SIAM J. Numer. Anal., 39(2001), 902931.CrossRefGoogle Scholar
[32]Schneider, R., Xu, Y. and Zhou, A., An analysis of discontinuous Galerkin methods for elliptic problems, Adv. Comput. Math., 25(2006), 259286.CrossRefGoogle Scholar
[33]Scott, L. R. and Zhang, S., Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp., 54(1994), 483493.CrossRefGoogle Scholar
[34]Tomar, S. K. and Repin, S. I., Efficient computable error bounds for discontinuous Galerkin pproximations of elliptic problems, J. Comput. Appl. Math., 226(2009), 358369.CrossRefGoogle Scholar
[35]Verfuürth, R., A Review of A Posteriori Estimation and Adaptive Mesh-Refinement Techniques, Wiley-Teubner, New York, Stuttgart, 1996.Google Scholar
[36]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. Comp., 68 (1999), 13471378.CrossRefGoogle Scholar
[37]Wohlmuth, B. I., Toselli, A. and Widlund, O. B., An iterative substructuring method for Raviart-Thomas vector fields in three dimensions, SIAM J. Numer. Anal., 37 (2000), 16571676.CrossRefGoogle Scholar

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A Priori and a Posteriori Error Estimates for H(div)-Elliptic Problem with Interior Penalty Method
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