This study focused on the superconvergence of the finite element method for the five-dimensional Poisson equation in the $ W^{1, \infty} $-seminorm. Specifically, we investigated the block finite element method, which is a tensor-product finite element approach applied to regular rectangular partitions of the domain. First, we introduced the finite element scheme for the equation and discussed various functions related to it, along with their properties. Next, we proposed a weight function and established its important properties, which play a crucial role in the theoretical analysis. By utilizing the properties of the weight function and employing weighted-norm analysis techniques, we derived an optimal order estimate in the $ W^{2, 1} $-seminorm for the discrete derivative Green's function (DDGF). Furthermore, we provided an interpolation fundamental estimate of the second type, also known as the weak estimate of the second type, for the block finite element. This weak estimate is based on a five-dimensional interpolation operator of the projection type. Finally, by combining the derived $ W^{2, 1} $-seminorm estimate for the DDGF and the weak estimate for the block finite element, we obtained a superconvergence estimate for the block finite element approximation in the pointwise sense of the $ W^{1, \infty} $-seminorm. The correctness of the theoretical results was demonstrated through a numerical example.
Citation: Jinghong Liu. $ W^{1, \infty} $-seminorm superconvergence of the block finite element method for the five-dimensional Poisson equation[J]. AIMS Mathematics, 2023, 8(12): 31092-31103. doi: 10.3934/math.20231591
This study focused on the superconvergence of the finite element method for the five-dimensional Poisson equation in the $ W^{1, \infty} $-seminorm. Specifically, we investigated the block finite element method, which is a tensor-product finite element approach applied to regular rectangular partitions of the domain. First, we introduced the finite element scheme for the equation and discussed various functions related to it, along with their properties. Next, we proposed a weight function and established its important properties, which play a crucial role in the theoretical analysis. By utilizing the properties of the weight function and employing weighted-norm analysis techniques, we derived an optimal order estimate in the $ W^{2, 1} $-seminorm for the discrete derivative Green's function (DDGF). Furthermore, we provided an interpolation fundamental estimate of the second type, also known as the weak estimate of the second type, for the block finite element. This weak estimate is based on a five-dimensional interpolation operator of the projection type. Finally, by combining the derived $ W^{2, 1} $-seminorm estimate for the DDGF and the weak estimate for the block finite element, we obtained a superconvergence estimate for the block finite element approximation in the pointwise sense of the $ W^{1, \infty} $-seminorm. The correctness of the theoretical results was demonstrated through a numerical example.
| [1] | C. M. Chen, Construction theory of superconvergence of finite elements (in Chinese), Changsha: Hunan Science and Technology Press, 2001. |
| [2] | C. M. Chen, Y. Q. Huang, High accuracy theory of finite element methods (in Chinese), Changsha: Hunan Science and Technology Press, 1995. |
| [3] | L. Wahlbin, Superconvergence in Galerkin finite element methods, Berlin: Springer Verlag, 1995. https://doi.org/10.1007/bfb0096835 |
| [4] | Q. D. Zhu, Q. Lin, Superconvergence theory of the finite element methods (in Chinese), Changsha: Hunan Science and Technology Press, 1989. |
| [5] | J. H. Brandts, M. Křížek, History and future of superconvergence in three dimensional finite element methods, In: Proceedings of the Conference on Finite Element Methods: Three-dimensional Problems, GAKUTO Int. Ser. Math. Sci. Appl., Gakkotosho, Tokyo, 15 (2001), 22–33. |
| [6] |
J. H. Brandts, M. Křížek, Gradient superconvergence on uniform simplicial partitions of polytopes, IMA J. Numer. Anal., 23 (2003), 489–505. https://doi.org/10.1093/imanum/23.3.489 doi: 10.1093/imanum/23.3.489
|
| [7] | J. H. 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. https://doi.org/10.1080/00207160601117354 doi: 10.1080/00207160601117354
|
| [10] |
G. Goodsell, Pointwise superconvergence of the gradient for the linear tetrahedral element, Numer. Meth. Part. D. E., 10 (1994), 651–666. https://doi.org/10.1002/num.1690100511 doi: 10.1002/num.1690100511
|
| [11] | V. Kantchev, R. D. Lazarov, Superconvergence of the gradient of linear finite elements for 3D Poisson equation, In: Proceedings of the Conference on Optimal Algorithms, Bulgarian Academy of Sciences, Sofia, 1986,172–182. |
| [12] |
J. H. Liu, B. Jia, Q. D. Zhu, An estimate for the three-dimensional discrete Green's function and applications, J. Math. Anal. Appl., 370 (2010), 350–363. https://doi.org/10.1016/j.jmaa.2010.05.002 doi: 10.1016/j.jmaa.2010.05.002
|
| [13] |
J. H. Liu, Q. D. Zhu, Maximum-norm superapproximation of the gradient for the trilinear block finite element, Numer. Meth. Part. D. E., 23 (2007), 1501–1508. https://doi.org/10.1002/num.20237 doi: 10.1002/num.20237
|
| [14] |
J. H. Liu, Q. D. Zhu, Pointwise supercloseness of tensor-product block finite elements, Numer. Meth. Part. D. E., 25 (2009), 990–1008. https://doi.org/10.1002/num.20384 doi: 10.1002/num.20384
|
| [15] | A. Pehlivanov, Superconvergence of the gradient for quadratic 3D simplex finite elements, In: Proceedings of the Conference on Numerical Methods and Application, Bulgarian Academy of Sciences, Sofia, 1989,362–366. |
| [16] |
A. H. Schatz, I. H. Sloan, L. B. Wahlbin, Superconvergence in finite element methods and meshes that are locally symmetric with respect to a point, SIAM J. Numer. Anal., 33 (1996), 505–521. https://doi.org/10.1137/0733027 doi: 10.1137/0733027
|
| [17] |
Z. M. Zhang, R. C. Lin, Locating natural superconvergent points of finite element methods in 3D, Int. J. Numer. Anal. Mod., 2 (2005), 19–30. https://doi.org/10.1080/00207160412331291107 doi: 10.1080/00207160412331291107
|
| [18] |
G. Goodsell, J. R. Whiteman, Pointwise superconvergence of recovered gradients for piecewise linear finite element approximations to problems of planar linear elasticity, Numer. Meth. Part. D. E., 6 (1990), 59–74. https://doi.org/10.1002/num.1690060105 doi: 10.1002/num.1690060105
|
| [19] | J. H. Liu, Q. D. Zhu, The $W^{1, 1}$-seminorm estimate for the discrete derivative Green's function for the 5D Poisson equation, J. Comput. Anal. Appl., 13 (2011), 1143–1156. |
| [20] |
G. H. Zhou, R. Rannacher, Pointwise superconvergence of the streamline diffusion finite-element method, Numer. Meth. Part. D. E., 12 (1996), 123–145. https://doi.org/10.1002/(sici)1098-2426(199601)12:1<123::aid-num7>3.0.co;2-u doi: 10.1002/(sici)1098-2426(199601)12:1<123::aid-num7>3.0.co;2-u
|
| [21] | R. A. Adams, Sobolev spaces, New York: Academic Press, 1975. |
Jinghong Liu. $ W^{1, \infty} $-seminorm superconvergence of the block finite element method for the five-dimensional Poisson equation[J]. AIMS Mathematics, 2023, 8(12): 31092-31103. doi: 10.3934/math.20231591
DownLoad: