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Enhancing finite element approximation for eigenvalue problems by projection method. (English) Zbl 1253.74107
Summary: We establish the superconvergence and the related recovery type a posteriori error estimators based on projection method for finite element approximation of the elliptic eigenvalue problems. The projection method is a postprocessing procedure that constructs a new approximation by using the least squares method. The results are based on some regularity assumption for the elliptic problem, and are applicable to the finite element approximations of self-adjoint elliptic eigenvalue problems with general quasi-regular partitions. Therefore, the result of this paper can be employed to provide useful a posteriori error estimators in adaptive finite element computation under unstructured meshes.

74S05 Finite element methods applied to problems in solid mechanics
74R99 Fracture and damage
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74K20 Plates
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
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[1] Adams, R.A., Sobolev spaces, (1975), Academic Press New York · Zbl 0186.19101
[2] Arnold, D.N.; Mukherjee, A.; Pouly, L., Locally adapted tetrahedral meshes using bisection, SIAM J. sci. comput., 22, 431-448, (2000) · Zbl 0973.65116
[3] Babuška, I.; Osborn, J.E., Finite element-Galerkin approximation for the eigenvalues and eigenvectors of selfadjoint problems, Math. comp., 52, 275-297, (1989) · Zbl 0675.65108
[4] Babuška, I.; Osborn, J.E., Eigenvalue problems, (), 641-792 · Zbl 0875.65087
[5] Becker, R.; Rannacher, R., An optimal control approach to a posteriori error estimation in finite element methods, Acta numerica, 10, 1-102, (2001) · Zbl 1105.65349
[6] Blum, H.; Rannacher, R., Finite element eigenvalue computation on domains with reentrant corners using Richardson extrapolation, J. comp. math., 8, 321-332, (1990) · Zbl 0719.65077
[7] Boffi, D., Finite element approximation of eigenvalue problems, Acta numerica, 19, 1-120, (2010) · Zbl 1242.65110
[8] Chen, H.; Wang, J., An interior estimate of superconvergence for finite element solutions for second-order elliptic problems on quasi-uniform meshes by local projections, SIAM. J. numer. anal., 41, 1318-1338, (2003) · Zbl 1058.65118
[9] Ciarlet, P.G., The finite element method for elliptic problems, (1978), North-Holland New York · Zbl 0445.73043
[10] Dai, X.; Xu, J.; Zhou, A., Convergence and optimal complexity of adaptive finite element eigenvalue computations, Numer. math., 110, 313-355, (2008) · Zbl 1159.65090
[11] Dörfler, W.; Nochetto, R.H., Small data oscillation implies the saturation assumption, Numer. math., 91, 1-12, (2002) · Zbl 0995.65109
[12] Durán, R.G.; Padra, C.; Rodriguez, R., A posteriori error estimators for the finite element approximations of eigenvalue problems, Math. models methods appl. sci., 13, 1219-1229, (2003) · Zbl 1072.65144
[13] Giani, S.; Graham, I.G., A convergence adaptive method for elliptic eigenvlue problems, SIAM J. numer. anal., 47, 1067-1091, (2009) · Zbl 1191.65147
[14] Heimsund, B.O.; Tai, X.C.; Wang, J.P., Superconvergence for the gradient of finite element approximations by \(L^2\) projections, SIAM J. numer. anal., 40, 1263-1280, (2002) · Zbl 1047.65095
[15] Heuveline, V.; Rannacher, R., A posteriori error control for finite element approximations of elliptic eigenvalue problems, Adv. comput. math., 15, 107-138, (2001) · Zbl 0995.65111
[16] Hoffmann, W.; Schatz, A.H.; Wahlbin, L.B.; Wittum, G., Asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes. part 1: A smooth problem and globally quasi-uniform meshes, Math. comp., 70, 897-909, (2001) · Zbl 0969.65099
[17] Huang, H.; Chang, S.; Chien, C.; Li, Z., Superconvergence of high order FEMs for eigenvalue problems with periodic boundary conditions, Comput. methods appl. mech. engrg., 198, 2246-2259, (2009) · Zbl 1229.81010
[18] Johnson, C., Numerical solution of partial differential equations by the finite element method, (1987), Cambridge University Press
[19] Larson, M.G., A posteriori and a priori error analysis for finite element approximations of self-adjoint elliptic eigenvalue problems, SIAM J. numer. anal., 38, 608-625, (2000) · Zbl 0974.65100
[20] Li, J.; Wang, J.; Ye, X., Superconvergence by \(L^2\)-projections for stabilized finite element methods for the Stokes equations, Int. J. numer. anal. model., 6, 711-723, (2009)
[21] Lin, Q.; Lin, J., Finite element methods, accuracy and improvements, (2006), Science Press Beijing
[22] Liu, H.; Sun, J., Recovery type a posteriori estimates and superconvergence for nonconforming FEM of eigenvalue problems, Appl. math. model., 33, 3488-3497, (2009) · Zbl 1205.65306
[23] Mao, D.; Shen, L.; Zhou, A., Adaptive finite element algorithms for eigenvalue problems based on local averaging type a posteriori error estimates, Adv. comput. math., 25, 135-160, (2006) · Zbl 1103.65112
[24] Naga, A.; Zhang, Z.; Zhou, A., Enhancing eigenvalue approximation by gradient recovery, SIAM J. sci. comput., 28, 1289-1300, (2006) · Zbl 1148.65087
[25] Naga, A.; Zhang, Z., A posteriori error estimates based on polynomial preserving recovery, SIAM J. numer. anal., 42, 1780-1800, (2004) · Zbl 1078.65098
[26] Zhang, Z.; Naga, A., A new finite element gradient recovery method: superconvergence property, SIAM J. sci. comput., 26, 1192-1213, (2005) · Zbl 1078.65110
[27] Schatz, A.H.; Wahlbin, L.B., Asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes. part II: the piecewise linear case, Math. comput., 73, 517-523, (2003) · Zbl 1038.65115
[28] Verfürth, R., A review of a posteriori error estimates and adaptive mesh-refinement techniques, (1996), Wiley-Teubner New York · Zbl 0853.65108
[29] Wang, J., A superconvergence analysis for finite element solutions by the least-squares surface Fitting on irregular meshes for smooth problems, J. math. study, 33, 229-243, (2000) · Zbl 0987.65108
[30] Wang, J.; Ye, X., Superconvergence of finite element approximations for the Stokes problem by projection methods, SIAM J. numer. anal., 39, 1001-1013, (2001) · Zbl 1002.65118
[31] Wu, H.; Zhang, Z., Enhancing eigenvalue approximation by gradient recovery on adaptive meshes, IMA J. numer. anal., 29, 1008-1022, (2009) · Zbl 1184.65101
[32] Xu, J.; Zhou, A., A two-grid discretization scheme for eigenvalue problems, Math. comput., 70, 17-25, (2001) · Zbl 0959.65119
[33] Xu, J.; Zhou, A., Local and parallel finite element algorithms for eigenvalue problems, Acta math. appl. sin., 18, 185-200, (2002) · Zbl 1015.65060
[34] Yang, Y., An analysis of the finite element method for eigenvalue problems, (2004), Guizhou Press Guiyang
[35] Yang, Y., A posteriori error analysis of conforming/nonconforming finite elements, Sci. sin. math., 40, 843-862, (2010), (in Chinese)
[36] Ye, X., Superconvergence of nonconforming finite element method for the Stokes equations, Numer. methods part. differ. eqs., 18, 143-154, (2002) · Zbl 1003.65121
[37] Zienkiewicz, O.C.; Zhu, J.Z., The superconvergence patch recovery and a posteriori error estimates, Int. J. numer. methods engrg., 33, 1331-1382, (1992) · Zbl 0769.73085
[38] Zienkiewicz, O.C.; Zhu, J.Z., The superconvergence patch recovery (SPR) and adaptive finite element refinement, Comput. methods appl. mech. engrg., 101, 207-224, (1992) · Zbl 0779.73078
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