This study investigated the point-wise superconvergence of block finite elements for the variable coefficient elliptic equation in a regular family of rectangular partitions of the domain in three-dimensional space. Initially, the estimates for the three-dimensional discrete Greens function and discrete derivative Greens function were presented. Subsequently, employing an interpolation operator of projection type, two essential weak estimates were derived, which were crucial for superconvergence analysis. Ultimately, by combining the aforementioned estimates, we achieved superconvergence estimates for the derivatives and function values of the finite element approximation in the point-wise sense of the $ L^\infty $-norm. A numerical example illustrated the theoretical results.
Citation: Jinghong Liu, Qiyong Li. Pointwise superconvergence of block finite elements for the three-dimensional variable coefficient elliptic equation[J]. AIMS Mathematics, 2024, 9(10): 28611-28622. doi: 10.3934/math.20241388
This study investigated the point-wise superconvergence of block finite elements for the variable coefficient elliptic equation in a regular family of rectangular partitions of the domain in three-dimensional space. Initially, the estimates for the three-dimensional discrete Greens function and discrete derivative Greens function were presented. Subsequently, employing an interpolation operator of projection type, two essential weak estimates were derived, which were crucial for superconvergence analysis. Ultimately, by combining the aforementioned estimates, we achieved superconvergence estimates for the derivatives and function values of the finite element approximation in the point-wise sense of the $ L^\infty $-norm. A numerical example illustrated the theoretical results.
| [1] | C. M. Chen, Y. Q. Huang, High accuracy theory of finite element methods (in Chinese), Changsha: Hunan Science and Technology Press, 1995. |
| [2] | P. G. Ciarlet, The finite element method for elliptic problems, Amsterdam: North-Holland, 1978. |
| [3] | J. H. Liu, Pointwise superconvergence analysis for multidimensional finite elements (in Chinese), Beijing: Science Press, 2019. |
| [4] | L. B. Wahlbin, Superconvergence in Galerkin finite element methods, Berlin: Springer Verlag, 1995. https://doi.org/10.1007/bfb0096835 |
| [5] | Q. D. Zhu, Q. Lin, Superconvergence theory of the finite element methods (in Chinese), Changsha: Hunan Science and Technology Press, 1989. |
| [6] | J. Brandts, M. Křížek, History and future of superconvergence in three dimensional finite element methods, Proceedings of the Conference on Finite Element Methods: Three-dimensional Problems, GAKUTO Int. Ser. Math. Sci. Appl., Gakkotosho, Tokyo, 15 (2001), 22–33. |
| [7] | J. Brandts, M. Křížek, Superconvergence of tetrahedral quadratic finite elements, J. Comput. Math., 23 (2005), 27–36. |
| [8] | C. M. Chen, Optimal points of the stresses for the linear tetrahedral element (in Chinese), Nat. Sci. J. Xiangtan Univ., 3 (1980), 16–24. |
| [9] | L. Chen, Superconvergence of tetrahedral linear finite elements, Int. J. Numer. Anal. Mod., 3 (2006), 273–282. |
| [10] |
G. Goodsell, Pointwise superconvergence of the gradient for the linear tetrahedral element, Numer. Methods Partial Differential Equations, 10 (1994), 651–666. https://doi.org/10.1002/num.1690100511 doi: 10.1002/num.1690100511
|
| [11] |
R. C. Lin, Z. M. Zhang, Natural superconvergence points in three-dimensional finite elements, SIAM J. Numer. Anal., 46 (2008), 1281–1297. https://doi.org/10.1137/070681168 doi: 10.1137/070681168
|
| [12] | J. H. Liu, Superconvergence of tensor-product quadratic pentahedral elements for variable coefficient elliptic equations, J. Comput. Anal. Appl., 14 (2012), 745–751. |
| [13] |
J. H. Liu, G. Hu, Q. D. Zhu, Superconvergence of tetrahedral quadratic finite elements for a variable coefficient elliptic equation, Numer. Methods Partial Differential Equations, 29 (2013), 1043–1055. https://doi.org/10.1002/num.21744 doi: 10.1002/num.21744
|
| [14] |
J. H. Liu, Q. D. Zhu, Pointwise supercloseness of tensor-product block finite elements, Numer. Methods Partial Differential Equations, 25 (2009), 990–1008. https://doi.org/10.1002/num.20384 doi: 10.1002/num.20384
|
| [15] |
J. H. Liu, Q. D. Zhu, Superconvergence of the function value for pentahedral finite elements for an elliptic equation with varying coefficients, Bound Value Probl., 2020 (2020), 1–15. https://doi.org/10.1186/s13661-019-01318-y doi: 10.1186/s13661-019-01318-y
|
Jinghong Liu, Qiyong Li. Pointwise superconvergence of block finite elements for the three-dimensional variable coefficient elliptic equation[J]. AIMS Mathematics, 2024, 9(10): 28611-28622. doi: 10.3934/math.20241388
DownLoad: