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

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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

Abstract
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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.

Keywords: PPR, discrete least-squares fitting, superconvergence, a posteriori error estimator., Symmetric interior penalty Galerkin method.

Mathematics Subject Classification: Primary: 65N15, 65N30, 97N5.

Citation: Lunji Song, Zhimin Zhang. Polynomial preserving recovery of an over-penalized symmetric interior penalty Galerkin method for elliptic problems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1405-1426. doi: 10.3934/dcdsb.2015.20.1405

References:

[1]	M. Ainsworth and J. T. Oden, A Posteriori Error Estimation in Finite Element Analysis,, Wiley Interscience, (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. 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. doi: 10.1002/nme.1620121010.
[4]	I. Babuška and T. Strouboulis, The Finite Element Method and Its Reliability,, Oxford University Press, (2001).
[5]	R. E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations,, Math. Comp., 44 (1985), 283. 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.
[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. 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. doi: 10.1090/S0025-5718-07-01951-5.
[9]	P. G. Ciarlet, Basic error estimates for elliptic problems,, in Handbook of Numerical Analysis, (1991), 17.
[10]	Y. Epshteyn and B. Rivière, Estimation of penalty parameters for symmetric interior penalty Galerkin methods,, J. Comput. Appl. Math., 206 (2007), 843. doi: 10.1016/j.cam.2006.08.029.
[11]	P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Monogr. Stud. Math. 24, (1985).
[12]	Y. Huang and J. Xu, Superconvergence of quadratic finite elements on mildly structured grids,, Math. Comp., 77 (2008), 1253. 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. doi: 10.3934/dcdsb.2005.5.769.
[14]	P. Oswald, On a BPX-preconditioner for $P1$ elements,, Computing, 51 (1993), 125. doi: 10.1007/BF02243847.
[15]	P. O. Persson and G. Strang, A simple mesh generator in Matlab,, SIAM Rev., 46 (2004), 329. doi: 10.1137/S0036144503429121.
[16]	B. Rivière, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation,, SIAM, (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. 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.
[19]	Z. Zhang and A. Naga, A new finite element gradient recovery method: Superconvergence property,, SIAM J. Sci. Comput., 26 (2005), 1192. 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. 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. 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. 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. 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. doi: 10.1090/S0025-5718-03-01600-4.

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References:

[1]	M. Ainsworth and J. T. Oden, A Posteriori Error Estimation in Finite Element Analysis,, Wiley Interscience, (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. 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. doi: 10.1002/nme.1620121010.
[4]	I. Babuška and T. Strouboulis, The Finite Element Method and Its Reliability,, Oxford University Press, (2001).
[5]	R. E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations,, Math. Comp., 44 (1985), 283. 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.
[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. 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. doi: 10.1090/S0025-5718-07-01951-5.
[9]	P. G. Ciarlet, Basic error estimates for elliptic problems,, in Handbook of Numerical Analysis, (1991), 17.
[10]	Y. Epshteyn and B. Rivière, Estimation of penalty parameters for symmetric interior penalty Galerkin methods,, J. Comput. Appl. Math., 206 (2007), 843. doi: 10.1016/j.cam.2006.08.029.
[11]	P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Monogr. Stud. Math. 24, (1985).
[12]	Y. Huang and J. Xu, Superconvergence of quadratic finite elements on mildly structured grids,, Math. Comp., 77 (2008), 1253. 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. doi: 10.3934/dcdsb.2005.5.769.
[14]	P. Oswald, On a BPX-preconditioner for $P1$ elements,, Computing, 51 (1993), 125. doi: 10.1007/BF02243847.
[15]	P. O. Persson and G. Strang, A simple mesh generator in Matlab,, SIAM Rev., 46 (2004), 329. doi: 10.1137/S0036144503429121.
[16]	B. Rivière, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation,, SIAM, (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. 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.
[19]	Z. Zhang and A. Naga, A new finite element gradient recovery method: Superconvergence property,, SIAM J. Sci. Comput., 26 (2005), 1192. 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. 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. 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. 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. 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. doi: 10.1090/S0025-5718-03-01600-4.

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