American Institute of Mathematical Sciences

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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

Lunji Song 1, and  Zhimin Zhang 2,

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.
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.  Google Scholar

[2]

D. N. Arnold, An interior penalty finite element method with discontinuous elements,, SIAM J. Numer. Anal., 19 (1982), 742.  doi: 10.1137/0719052.  Google Scholar

[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.  Google Scholar

[4]

I. Babuška and T. Strouboulis, The Finite Element Method and Its Reliability,, Oxford University Press, (2001).   Google Scholar

[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.  Google Scholar

[6]

C. Brenner, L. Owens and L.-Y. Sung, A weakly over-penalized symmetric interior penalty method,, Electron. Trans. Numer. Anal., 30 (2008), 107.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

P. G. Ciarlet, Basic error estimates for elliptic problems,, in Handbook of Numerical Analysis, (1991), 17.   Google Scholar

[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.  Google Scholar

[11]

P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Monogr. Stud. Math. 24, (1985).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

P. Oswald, On a BPX-preconditioner for $P1$ elements,, Computing, 51 (1993), 125.  doi: 10.1007/BF02243847.  Google Scholar

[15]

P. O. Persson and G. Strang, A simple mesh generator in Matlab,, SIAM Rev., 46 (2004), 329.  doi: 10.1137/S0036144503429121.  Google Scholar

[16]

B. Rivière, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation,, SIAM, (2008).  doi: 10.1137/1.9780898717440.  Google Scholar

[17]

M. F. Wheeler, An elliptic collocation-finite element method with interior penalties,, SIAM J. Numer. Anal., 15 (1978), 152.  doi: 10.1137/0715010.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

M. Ainsworth and J. T. Oden, A Posteriori Error Estimation in Finite Element Analysis,, Wiley Interscience, (2000).  doi: 10.1002/9781118032824.  Google Scholar

[2]

D. N. Arnold, An interior penalty finite element method with discontinuous elements,, SIAM J. Numer. Anal., 19 (1982), 742.  doi: 10.1137/0719052.  Google Scholar

[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.  Google Scholar

[4]

I. Babuška and T. Strouboulis, The Finite Element Method and Its Reliability,, Oxford University Press, (2001).   Google Scholar

[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.  Google Scholar

[6]

C. Brenner, L. Owens and L.-Y. Sung, A weakly over-penalized symmetric interior penalty method,, Electron. Trans. Numer. Anal., 30 (2008), 107.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

P. G. Ciarlet, Basic error estimates for elliptic problems,, in Handbook of Numerical Analysis, (1991), 17.   Google Scholar

[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.  Google Scholar

[11]

P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Monogr. Stud. Math. 24, (1985).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

P. Oswald, On a BPX-preconditioner for $P1$ elements,, Computing, 51 (1993), 125.  doi: 10.1007/BF02243847.  Google Scholar

[15]

P. O. Persson and G. Strang, A simple mesh generator in Matlab,, SIAM Rev., 46 (2004), 329.  doi: 10.1137/S0036144503429121.  Google Scholar

[16]

B. Rivière, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation,, SIAM, (2008).  doi: 10.1137/1.9780898717440.  Google Scholar

[17]

M. F. Wheeler, An elliptic collocation-finite element method with interior penalties,, SIAM J. Numer. Anal., 15 (1978), 152.  doi: 10.1137/0715010.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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