# American Institute of Mathematical Sciences

July  2015, 20(5): 1405-1426. doi: 10.3934/dcdsb.2015.20.1405

## Polynomial preserving recovery of an over-penalized symmetric interior penalty Galerkin method for elliptic problems

 1 School of Mathematics and Statistics, and Key Laboratory of Applied Mathematics and Complex Systems in Gansu Province, Lanzhou University, Lanzhou 730000, China 2 Beijing Computational Science Research Center, Beijing 100094, China

Received  February 2014 Revised  January 2015 Published  May 2015

A polynomial preserving recovery technique is applied to an over-penalized symmetric interior penalty method. The discontinuous Galerkin solution values are used to recover the gradient and to further construct an a posteriori error estimator in the energy norm. In addition, for uniform triangular meshes and mildly structured meshes satisfying the $\epsilon$-$\sigma$ condition, the method for the linear element is superconvergent under the regular pattern and under the chevron pattern, while it is superconvergent for the quadratic element under the regular pattern.
Citation: Lunji Song, Zhimin Zhang. Polynomial preserving recovery of an over-penalized symmetric interior penalty Galerkin method for elliptic problems. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1405-1426. doi: 10.3934/dcdsb.2015.20.1405
##### References:
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##### References:
 [1] M. Ainsworth and J. T. Oden, A Posteriori Error Estimation in Finite Element Analysis, Wiley Interscience, New York, 2000. doi: 10.1002/9781118032824. [2] D. N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal., 19 (1982), 742-760. doi: 10.1137/0719052. [3] I. Babuška and W. C. Rheinboldt, A-Posteriori Error Estimates for the Finite Element Method, Internat. J. Numer. Methods Engrg., 12 (1978), 1597-1615. doi: 10.1002/nme.1620121010. [4] I. Babuška and T. Strouboulis, The Finite Element Method and Its Reliability, Oxford University Press, London, 2001. [5] R. E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Math. Comp., 44 (1985), 283-301. doi: 10.1090/S0025-5718-1985-0777265-X. [6] C. Brenner, L. Owens and L.-Y. Sung, A weakly over-penalized symmetric interior penalty method, Electron. Trans. Numer. Anal., 30 (2008), 107-127. [7] S. C. Brenner, T. Gudi and L.-Y. Sung, A posteriori error control for a weakly over-penalized symmetric interior penalty method, J. Sci. Comput., 40 (2009), 37-50. doi: 10.1007/s10915-009-9278-0. [8] E. Burman and A. Ern, Continuous interior penalty hp-finite element methods for advection and advection-diffusion equations, Math. Comp., 76 (2007), 1119-1140. doi: 10.1090/S0025-5718-07-01951-5. [9] P. G. Ciarlet, Basic error estimates for elliptic problems, in Handbook of Numerical Analysis, Vol. II (eds. P.G. Ciarlet and J.L. Lions), North-Holland, Amsterdam, (1991), 17-351. [10] Y. Epshteyn and B. Rivière, Estimation of penalty parameters for symmetric interior penalty Galerkin methods, J. Comput. Appl. Math., 206 (2007), 843-872. doi: 10.1016/j.cam.2006.08.029. [11] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monogr. Stud. Math. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. [12] Y. Huang and J. Xu, Superconvergence of quadratic finite elements on mildly structured grids, Math. Comp., 77 (2008), 1253-1268. doi: 10.1090/S0025-5718-08-02051-6. [13] A. Naga and Z. Zhang, The polynomial-preserving recovery for higher order finite element methods in 2D and 3D, Discrete Continuous Dynam. Systems - B, 5 (2005), 769-798. doi: 10.3934/dcdsb.2005.5.769. [14] P. Oswald, On a BPX-preconditioner for $P1$ elements, Computing, 51 (1993), 125-133. doi: 10.1007/BF02243847. [15] P. O. Persson and G. Strang, A simple mesh generator in Matlab, SIAM Rev., 46 (2004), 329-345. doi: 10.1137/S0036144503429121. [16] B. Rivière, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation, SIAM, Philadelphia, PA, 2008. doi: 10.1137/1.9780898717440. [17] M. F. Wheeler, An elliptic collocation-finite element method with interior penalties, SIAM J. Numer. Anal., 15 (1978), 152-161. doi: 10.1137/0715010. [18] Z. Zhang, Polynomial preserving gradient recovery and a posteriori estimate for bilinear element on irregular quadrilaterals, Int. J. Num. Anal. Model., 1 (2004), 1-24. [19] Z. Zhang and A. Naga, A new finite element gradient recovery method: Superconvergence property, SIAM J. Sci. Comput., 26 (2005), 1192-1213. doi: 10.1137/S1064827503402837. [20] Z. Zhang and A. Naga, A posteriori error estimates based on polynomial preserving recovery, SIAM J. Numer. Anal., 42 (2004), 1780-1800. doi: 10.1137/S0036142903413002. [21] Z. Zhang, Polynomial preserving recovery for meshes from Delaunay triangulation or with high aspect ratio, Numer. Methods Partial Differential Equations, 24 (2008), 960-971. doi: 10.1002/num.20300. [22] O. C. Zienkiewicz and J. Z. Zhu, A simple error estimator and adaptive procedure for practical engineering analysis, Internat. J. Numer. Methods Engrg., 24 (1987), 337-357. doi: 10.1002/nme.1620240206. [23] O. C. Zienkiewicz and J. Z. Zhu, The superconvergent patch recovery and a posteriori error estimates, Part 1: The recovery technique, Internat. J. Numer. Methods Engrg., 33 (1992), 1331-1364. doi: 10.1002/nme.1620330702. [24] J. Xu and Z. Zhang, Analysis of recovery type a posteriori error estimators for mildly structured grids, Math. Comp., 73 (2004), 1139-1152. doi: 10.1090/S0025-5718-03-01600-4.
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