Superconvergence of quadratic finite volume method on triangular meshes. (English) Zbl 1404.65229
Summary: This paper is concerned with the superconvergence properties of the quadratic finite volume method (FVM) on triangular meshes for elliptic equations. We proved the 3rd order superconvergence rate of the gradient approximation \(\| u_h - u_I \|_1 = O(h^3)\) and the 4th order superconvergence rate of the function value approximation \(\| u_h - u_I \|_0 = O(h^4)\) for the quadratic FVM on triangular meshes. Here \(u_h\) is the FVM solution and \(u_I\) is the piecewise quadratic Lagrange interpolation of the exact solution. It should be pointed out that the superconvergence phenomena of FVM strongly depends on the construction of dual mesh. Specially for quadratic FVMs, the scheme presented in this paper is the unique scheme which holds superconvergence.
MSC:
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 65N08 | Finite volume methods for boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
References:
| [1] | (Zienkiewicz, O. C., The Finite Element Method in Structural and Continuum Mechanics: Numerical Solution of Problems in Structural and Continuum Mechanics, European Civil Eng. Ser., vol. 1, (1967), McGraw-Hill: McGraw-Hill London) · Zbl 0189.24902 |
| [2] | Andreev, A. B.; Lazarov, R. D., Superconvergence of the gradient for quadratic triangular finite elements, Numer. Methods Partial Differential Equations, 4, 1, 15-32, (1988) · Zbl 0644.65082 |
| [3] | Babuška, I.; Banerjee, U.; Osborn, J. E., Superconvergence in the generalized finite element method, Numer. Math., 107, 3, 353-395, (2007) · Zbl 1129.65075 |
| [4] | Bank, R. E.; Xu, J., Asymptotically exact a posteriori error estimators, part i: grids with superconvergence, SIAM J. Numer. Anal., 41, 6, 2294-2312, (2003) · Zbl 1058.65116 |
| [5] | Chen, C.; Hu, S., The highest order superconvergence for bi-\(k\) degree rectangular elements at nodes: A proof of \(2 k\)-conjecture, Math. Comp., 82, 283, 1337-1355, (2013) · Zbl 1276.65067 |
| [6] | Chen, C.; Huang, Y., High Accuracy Theory of Finite Element Methods (In Chinese), (1995), Hunan Science and Technology Publishing House: Hunan Science and Technology Publishing House Changsha, China. |
| [7] | Křížek, M.; Neittaanmäki, P., On superconvergence techniques, Acta Appl. Math., 9, 3, 175-198, (1987) · Zbl 0624.65107 |
| [8] | Lin, Q.; Yan, N., The Construction and Analysis for Efficient Finite Elements (In Chinese), (1996), Hebei Univ. Publ. House |
| [9] | Schatz, A. H.; Sloan, I. H.; Wahlbin, L. B., Superconvergence in finite element methods and meshes that are locally symmetric with respect to a point, SIAM J. Numer. Anal., 33, 2, 505-521, (1996) · Zbl 0855.65115 |
| [10] | Thomée, V., High order local approximations to derivatives in the finite element method, Math. Comp., 31, 139, 652-660, (1977) · Zbl 0367.65055 |
| [11] | (Wahlbin, L. B., Superconvergence in Galerkin Finite Element Methods, Lecture Notes in Mathematics, vol. 1605, (1995), Springer: Springer Berlin) · Zbl 0826.65092 |
| [12] | Wang, J.; Ye, X., Superconvergence of finite element approximations for the stokes problem by projection methods, SIAM J. Numer. Anal., 39, 3, 1001-1013, (2001) · Zbl 1002.65118 |
| [13] | Zhu, Q., Superconvergent consistent estimates of the finite element (In Chinese), J. Xiangtan Univ., 1, 10-26, (1985) · Zbl 0618.65092 |
| [14] | Zhu, Q.; Lin, Q., The superconvergence theory of finite elements (In Chinese), (1989), Hunan Science and Technology Publishing House: Hunan Science and Technology Publishing House Changsha, China |
| [15] | Bank, R. E.; Rose, D. J., Some error estimates for the box method, SIAM J. Numer. Anal., 24, 4, 777-787, (1987) · Zbl 0634.65105 |
| [16] | Barth, T.; Ohlberger, M., Finite Volume Methods: Foundation and Analysis, (2004), Wiley: Wiley Chichester |
| [17] | Cai, Z., On the finite volume element method, Numer. Math., 58, 1, 713-735, (1990) · Zbl 0731.65093 |
| [18] | Cai, Z.; Mandel, J.; McCormick, S., The Finite volume element method for diffusion equations on general triangulations, SIAM J. Numer. Anal., 28, 2, 392-402, (1991) · Zbl 0729.65086 |
| [19] | Cao, W.; Zhang, Z.; Zou, Q., Superconvergence of any order finite volume schemes for 1d general elliptic equations, J. Sci. Comput., 56, 3, 566-590, (2013) · Zbl 1276.65044 |
| [20] | Chen, L., A new class of high order finite volume methods for second order elliptic equations, SIAM J. Numer. Anal., 47, 6, 4021-4043, (2010) · Zbl 1261.65109 |
| [21] | Chen, Z.; Li, R.; Zhou, A., A note on the optimal l2-estimate of the finite volume element method, Adv. Comput. Math., 16, 4, 291-303, (2002) · Zbl 0997.65122 |
| [22] | Chen, Z.; Wu, J.; Xu, Y., Higher-order finite volume methods for elliptic boundary value problems, Adv. Comput. Math., 37, 2, 191-253, (2012) · Zbl 1266.65180 |
| [23] | Chen, Z.; Xu, Y.; Zhang, Y., Higher-order finite volume methods II: Inf-sup condition and uniform local ellipticity, J. Comput. Appl. Math., 265, 96-109, (2014) · Zbl 1293.65142 |
| [24] | Chou, S.-H.; Ye, X., Unified Analysis of Finite Volume Methods for Second Order Elliptic Problems, SIAM J. Numer. Anal., 45, 4, 1639-1653, (2007) · Zbl 1155.65099 |
| [25] | Hackbusch, W., On first and second order box schemes, Computing, 41, 4, 277-296, (1989) · Zbl 0649.65052 |
| [26] | Huang, J.; Xi, S., On the Finite Volume Element Method for General Self-Adjoint Elliptic Problems, SIAM J. Numer. Anal., 35, 5, 1762-1774, (1998) · Zbl 0913.65097 |
| [27] | Li, R.; Chen, Z.; Wu, W., Generalized Difference Methods for Differential Equations, (2000), Marcel Dekker: Marcel Dekker New York · Zbl 0940.65125 |
| [28] | Liebau, F., The finite volume element method with quadratic basis functions, Computing, 57, 4, 281-299, (1996) · Zbl 0866.65074 |
| [29] | Schmidt, T., Box schemes on quadrilateral meshes, Computing, 51, 3-4, 271-292, (1993) · Zbl 0787.65075 |
| [30] | Wu, H.; Li, R., Error estimates for finite volume element methods for general second-order elliptic problems, Numer. Methods Partial Differential Equations, 19, 6, 693-708, (2003) · Zbl 1040.65091 |
| [31] | Xu, J.; Zou, Q., Analysis of linear and quadratic simplicial finite volume methods for elliptic equations, Numer. Math., 111, 3, 469-492, (2009) · Zbl 1169.65110 |
| [32] | Cao, W.; Zhang, Z.; Zou, Q., Is 2k-conjecture valid for finite volume methods?, SIAM J. Numer. Anal., 53, 2, 942-962, (2015) · Zbl 1327.65216 |
| [33] | Lin, T.; Ye, X., A posteriori error estimates for finite volume method based on bilinear trial functions for the elliptic equation, J. Comput. Appl. Math., 254, 185-191, (2013) · Zbl 1290.65101 |
| [34] | Lin, Y.; Yang, M.; Zou, Q., L\({}^2\) error estimates for a class of any order finite volume schemes over quadrilateral meshes, SIAM J. Numer. Anal., 53, 4, 2030-2050, (2015) · Zbl 1327.65208 |
| [35] | Lv, J.; Li, Y., L\({}^2\) error estimates and superconvergence of the finite volume element methods on quadrilateral meshes, Adv. Comput. Math., 37, 3, 393-416, (2012) · Zbl 1255.65198 |
| [36] | Lv, J.; Li, Y., Optimal biquadratic finite volume element methods on quadrilateral meshes, SIAM J. Numer. Anal., 50, 5, 2379-2399, (2012) · Zbl 1263.65117 |
| [37] | Shu, S.; Yu, H.; Huang, Y.; Nie, C., A symmetric finite volume element scheme on quadrilateral grid and superconvergence, Int. J. Numer. Anal. Model., 3, 3, 348-360, (2006) · Zbl 1100.65102 |
| [38] | Wang, T.; Gu, Y., Superconvergent biquadratic finite volume element method for two-dimensional Poisson’s equations, J. Comput. Appl. Math., 234, 2, 447-460, (2010) · Zbl 1191.65143 |
| [39] | Yang, M., A second-order finite volume element method on quadrilateral meshes for elliptic equations, ESAIM Math. Model. Numer. Anal., 40, 6, 1053-1067, (2006) · Zbl 1141.65081 |
| [40] | Yang, M.; Liu, J.; Lin, Y., Quadratic finite-volume methods for elliptic and parabolic problems on quadrilateral meshes: optimal-order errors based on Barlow points, IMA J. Numer. Anal., 33, 4, 1342-1364, (2013) · Zbl 1281.65140 |
| [41] | Zhang, Z.; Zou, Q., Vertex-centered finite volume schemes of any order over quadrilateral meshes for elliptic boundary value problems, Numer. Math., 130, 2, 363-393, (2015) · Zbl 1338.65233 |
| [42] | Chen, Z., Superconvergence of generalized difference methods for elliptic boundary value problem, Numer. Math. J. Chin. Univ. (English Ser.), 3, 163-171, (1994) · Zbl 0814.65102 |
| [43] | Chou, S. H.; Ye, X., Superconvergence of finite volume methods for the second order elliptic problem, Comput. Methods Appl. Mech. Engrg., 196, 37-40, 3706-3712, (2007) · Zbl 1173.65354 |
| [44] | Ewing, R. E.; Lin, T.; Lin, Y., On the Accuracy of the Finite Volume Element Method Based on Piecewise Linear Polynomials, SIAM J. Numer. Anal., 39, 6, 1865-1888, (2002) · Zbl 1036.65084 |
| [45] | Li, R.; Zhu, P., Generalized difference methods for second order elliptic partial differential equations (I)-triangle grids (In Chinese), Numer. Math. J. Chinese Univ., 2, 140-152, (1982) · Zbl 0584.65064 |
| [46] | Wang, X.; Li, Y., L\({}^2\) Error Estimates for High Order Finite Volume Methods on Triangular Meshes, SIAM J. Numer. Anal., 54, 5, 2729-2749, (2016) · Zbl 1348.65155 |
| [47] | Zhang, T., Superconvergence of finite volume element method for elliptic problems, Adv. Comput. Math., 40, 2, 399-413, (2014) · Zbl 1298.65162 |
| [48] | Tian, M.; Chen, Z., A generalized difference method with quadratic elements for elliptic equations (In Chinese), Numer. Math. J. Chinese Univ., 13, 99-113, (1991) · Zbl 0734.65083 |
| [49] | Chen, Z.; Xu, Y.; Zhang, Y., A construction of higher-order finite volume methods, Math. Comp., 84, 292, 599-628, (2015) · Zbl 1307.65144 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
