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Volume 17, Issue 4
A Nonconforming P2 and Discontinuous P1 Mixed Finite Element on Tetrahedral Grids

Shangyou Zhang

Adv. Appl. Math. Mech., 17 (2025), pp. 1259-1274.

Published online: 2025-05

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  • Abstract

A nonconforming $P_2$ finite element is constructed by enriching the conforming $P_2$ finite element space with seven $P_2$ nonconforming bubble functions (out of fifteen such bubble functions on each tetrahedron). This spacial nonconforming $P_2$ finite element, combined with the discontinuous $P_1$ finite element on general tetrahedral grids, is inf-sup stable for solving the Stokes equations. Consequently, such a mixed finite element method produces optimal-order convergent solutions for solving the stationary Stokes equations. Numerical tests confirm the theory.

  • AMS Subject Headings

65N15, 65N30, 76M10

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-17-1259, author = {Zhang , Shangyou}, title = {A Nonconforming P2 and Discontinuous P1 Mixed Finite Element on Tetrahedral Grids}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2025}, volume = {17}, number = {4}, pages = {1259--1274}, abstract = {

A nonconforming $P_2$ finite element is constructed by enriching the conforming $P_2$ finite element space with seven $P_2$ nonconforming bubble functions (out of fifteen such bubble functions on each tetrahedron). This spacial nonconforming $P_2$ finite element, combined with the discontinuous $P_1$ finite element on general tetrahedral grids, is inf-sup stable for solving the Stokes equations. Consequently, such a mixed finite element method produces optimal-order convergent solutions for solving the stationary Stokes equations. Numerical tests confirm the theory.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2023-0316}, url = {http://global-sci.org/intro/article_detail/aamm/24061.html} }
TY - JOUR T1 - A Nonconforming P2 and Discontinuous P1 Mixed Finite Element on Tetrahedral Grids AU - Zhang , Shangyou JO - Advances in Applied Mathematics and Mechanics VL - 4 SP - 1259 EP - 1274 PY - 2025 DA - 2025/05 SN - 17 DO - http://doi.org/10.4208/aamm.OA-2023-0316 UR - https://global-sci.org/intro/article_detail/aamm/24061.html KW - Quadratic finite element, nonconforming finite element, mixed finite element, Stokes equations, tetrahedral grid. AB -

A nonconforming $P_2$ finite element is constructed by enriching the conforming $P_2$ finite element space with seven $P_2$ nonconforming bubble functions (out of fifteen such bubble functions on each tetrahedron). This spacial nonconforming $P_2$ finite element, combined with the discontinuous $P_1$ finite element on general tetrahedral grids, is inf-sup stable for solving the Stokes equations. Consequently, such a mixed finite element method produces optimal-order convergent solutions for solving the stationary Stokes equations. Numerical tests confirm the theory.

Zhang , Shangyou. (2025). A Nonconforming P2 and Discontinuous P1 Mixed Finite Element on Tetrahedral Grids. Advances in Applied Mathematics and Mechanics. 17 (4). 1259-1274. doi:10.4208/aamm.OA-2023-0316
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