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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 |
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. |
show all references
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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