Interpolated Galerkin Finite Elements on Rectangular and Cuboid Meshes for the Biharmonic Equation

doi:10.1007/s42967-024-00473-z

Communications on Applied Mathematics and Computation ›› 2026, Vol. 8 ›› Issue (2): 765-777.doi: 10.1007/s42967-024-00473-z

Interpolated Galerkin Finite Elements on Rectangular and Cuboid Meshes for the Biharmonic Equation

Mengjiao Pan¹, Tatyana Sorokina², Shangyou Zhang³

1. School of Science, East China Jiaotong University, Nanchang, 330013, Jiangxi, China;
2. Department of Mathematics, Towson University, Towson, MD, 21252, USA;
3. Department of Mathematical Sciences, University of Delaware, Newark, DE, 19716, USA

收稿日期:2024-11-28 修回日期:2024-12-15 出版日期:2026-04-07 发布日期:2026-04-07
通讯作者: Shangyou Zhang,E-mail:szhang@udel.edu E-mail:szhang@udel.edu

Interpolated Galerkin Finite Elements on Rectangular and Cuboid Meshes for the Biharmonic Equation

Mengjiao Pan¹, Tatyana Sorokina², Shangyou Zhang³

1. School of Science, East China Jiaotong University, Nanchang, 330013, Jiangxi, China;
2. Department of Mathematics, Towson University, Towson, MD, 21252, USA;
3. Department of Mathematical Sciences, University of Delaware, Newark, DE, 19716, USA

Received:2024-11-28 Revised:2024-12-15 Online:2026-04-07 Published:2026-04-07
Contact: Shangyou Zhang,E-mail:szhang@udel.edu E-mail:szhang@udel.edu

摘要/Abstract

摘要： A new Galerkin finite element for the biharmonic equation is constructed on 2D rectangular and 3D cuboid meshes. In this \begin{document}$ C^1 $\end{document}-\begin{document}$ Q_k $\end{document} (\begin{document}$ k\geqslant 4 $\end{document}) interpolated Galerkin finite element construction, all unknowns associated with the interior of each element are determined by the direct interpolation of the right-hand-side function, and the rest of the unknowns, associated with the boundary of each element, are determined by solving the remaining linear equations of the Galerkin projection. In comparison with the traditional finite element method in two dimensions, our method reduces the number of unknowns from \begin{document}$ O(k^2) $\end{document} to O(k). Additionally, the method reduces the condition number drastically as it requires only 1% of the computer time, compared with the standard finite elements, in several numerical tests. We prove the existence and uniqueness of the solution and the optimal order of convergence. We confirm the theory by numerical tests in two dimensions and three dimensions.

关键词: Finite element, Interpolated finite element, Rectangular grid, Biharmonic equation, Tensor product

Abstract: A new Galerkin finite element for the biharmonic equation is constructed on 2D rectangular and 3D cuboid meshes. In this \begin{document}$ C^1 $\end{document}-\begin{document}$ Q_k $\end{document} (\begin{document}$ k\geqslant 4 $\end{document}) interpolated Galerkin finite element construction, all unknowns associated with the interior of each element are determined by the direct interpolation of the right-hand-side function, and the rest of the unknowns, associated with the boundary of each element, are determined by solving the remaining linear equations of the Galerkin projection. In comparison with the traditional finite element method in two dimensions, our method reduces the number of unknowns from \begin{document}$ O(k^2) $\end{document} to O(k). Additionally, the method reduces the condition number drastically as it requires only 1% of the computer time, compared with the standard finite elements, in several numerical tests. We prove the existence and uniqueness of the solution and the optimal order of convergence. We confirm the theory by numerical tests in two dimensions and three dimensions.

Key words: Finite element, Interpolated finite element, Rectangular grid, Biharmonic equation, Tensor product

中图分类号:

Mengjiao Pan, Tatyana Sorokina, Shangyou Zhang. Interpolated Galerkin Finite Elements on Rectangular and Cuboid Meshes for the Biharmonic Equation[J]. Communications on Applied Mathematics and Computation, 2026, 8(2): 765-777.

参考文献

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[4] Huang, Y., Zhang, S.: Supercloseness of the divergence-free finite element solutions on rectangular grids. Commun. Math. Stat. 1(2), 143–162 (2013)
[5] Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54, 483–493 (1990)
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[7] Sorokina, T., Zhang, S.: Conforming harmonic finite elements on the Hsieh-Clough-Tocher split of a triangle. Int. J. Numer. Anal. Model. 17(1), 54–67 (2020)
[8] Sorokina, T., Zhang, S.: Conforming and nonconforming harmonic finite elements. Appl. Anal. 99(4), 569–584 (2020)
[9] Sorokina, T., Zhang, S.: An interpolated Galerkin finite element method for the Poisson equation. J. Sci. Comput. 92(2), 47 (2022)
[10] Sorokina, T., Zhang, S. Neamtu, M., Fasshauer, G. E., Schumaker, L. L., Neamtu, M., Schumaker, L. L., Fasshauer, G. E.: Trivariate Interpolated Galerkin Finite Elements for the Poisson Equation, Approximation Theory XVI, pp. 237–249, Springer Proc. Math. Stat., 336, Springer, Cham (2021)
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Interpolated Galerkin Finite Elements on Rectangular and Cuboid Meshes for the Biharmonic Equation

Interpolated Galerkin Finite Elements on Rectangular and Cuboid Meshes for the Biharmonic Equation

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