Abstract
When applying finite element method to the Poisson equation on a domain in \(\mathbb {R}^3\), we replace some Lagrange nodal basis functions by bubble functions whose dual functionals are the values of the Laplacian. To compute the coefficients of these Laplacian basis functions instead of solving a large linear system, we interpolate the right hand side function in the Poisson equation. The finite element solution is then the Galerkin projection on a smaller vector space. We construct a qudratic and a cubic nonconforming interpolated finite elements, and quartic and higher degree conforming interpolated finite elements on arbitrary tetrahedral partitions. The main advantage of our method is that the number of unknowns that require solving a large system of equations on each element is reduced. We show that the interpolated Galerkin finite element method retains the optimal order of convergence. Numerical results confirming the theory are provided as well as comparisons with the standard finite elements.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Alfeld, P., Sorokina T.: Linear differential operators on bivariate spline spaces and spline vector fields. BIT Numer. Math. 56(1), 15–32 (2016)
Arnold, D.N., Boffi, D., Falk, R.S.: Approximation by quadrilateral finite elements. Math. Comput. 71(239), 909–922 (2002)
Brenner, S.C., Scott, L.R.: The mathematical theory of finite element methods. In: Texts in Applied Mathematics, vol. 15. Springer, New York (2008)
Falk, R.S., Gatto P., Monk, P.: Hexahedral H(div) and H(curl) finite elements. ESAIM Math. Model. Numer. Anal. 45(1), 115–143 (2011)
Fortin, M.: A three-dimensional quadratic nonconforming element. Numer. Math. 46, 269–279 (1985)
Hu, J., Huang Y., Zhang S.: The lowest order differentiable finite element on rectangular grids. SIAM Numer. Anal. 49(4), 1350–1368 (2011)
Hu, J., Zhang, S.: The minimal conforming H k finite element spaces on R n rectangular grids. Math. Comput. 84(292), 563–579 (2015)
Hu, J., Zhang, S.: Finite element approximations of symmetric tensors on simplicial grids in R n: the lower order case. Math. Models Methods Appl. Sci. 26(9), 1649–1669 (2016)
Huang, Y., Zhang, S.: Supercloseness of the divergence-free finite element solutions on rectangular grids. Commun. Math. Stat. 1(2), 143–162 (2013)
Schumaker, L.L., Sorokina, T., Worsey, A.J.: A C1 quadratic trivariate macro-element space defined over arbitrary tetrahedral partitions. J. Approx. Theory 158(1), 126–142 (2009)
Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54, 483–493 (1990)
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)
Sorokina, T., Zhang, S.: Conforming and nonconforming harmonic finite elements. Appl. Anal. https://doi.org/10.1080/00036811.2018.1504031
Sorokina, T., Zhang, S.: An interpolated Galerkin finite element method for the Poisson equation (preprint)
Zhang, S.: A C1-P2 finite element without nodal basis. Math. Model. Numer. Anal. 42, 175–192 (2008)
Zhang, S.: A family of 3D continuously differentiable finite elements on tetrahedral grids. Appl. Numer. Math. 59(1), 219–233 (2009)
Zhang, S.: A family of differentiable finite elements on simplicial grids in four space dimensions (Chinese). Math. Numer. Sin. 38(3), 309–324 (2016)
Zhang, S.: A P4 bubble enriched P3 divergence-free finite element on triangular grids. Comput. Math. Appl. 74(11), 2710–2722 (2017)
Acknowledgement
The first author is partially supported by a grant from the Simons Foundation #235411 to Tatyana Sorokina.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Sorokina, T., Zhang, S. (2021). Trivariate Interpolated Galerkin Finite Elements for the Poisson Equation. In: Fasshauer, G.E., Neamtu, M., Schumaker, L.L. (eds) Approximation Theory XVI. AT 2019. Springer Proceedings in Mathematics & Statistics, vol 336. Springer, Cham. https://doi.org/10.1007/978-3-030-57464-2_13
Download citation
DOI: https://doi.org/10.1007/978-3-030-57464-2_13
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-57463-5
Online ISBN: 978-3-030-57464-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)