Nonconforming Finite Element Method Applied to the Driven Cavity Problem

Lim, Roktaek; Sheen, Dongwoo

doi:10.4208/cicp.oa-2016-0039

Cited by 6 publications

(3 citation statements)

References 35 publications

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“…Recently, the proposed elements are used to solve a driven cavity problem [17] and an interface problem governed by the Stokes, Darcy, and Brinkman equations [16].…”

Section: In §5mentioning

confidence: 99%

Stable cheapest nonconforming finite elements for the Stokes equations

Kim

Yim

Sheen

2016

Journal of Computational and Applied Mathematics

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We introduce two pairs of stable cheapest nonconforming finite element space pairs to approximate the Stokes equations. One pair has each component of its velocity field to be approximated by the $P_1$ nonconforming quadrilateral element while the pressure field is approximated by the piecewise constant function with globally two-dimensional subspaces removed: one removed space is due to the integral mean--zero property and the other space consists of global checker--board patterns. The other pair consists of the velocity space as the $P_1$ nonconforming quadrilateral element enriched by a globally one--dimensional macro bubble function space based on $DSSY$ (Douglas-Santos-Sheen-Ye) nonconforming finite element space; the pressure field is approximated by the piecewise constant function with mean--zero space eliminated. We show that two element pairs satisfy the discrete inf-sup condition uniformly. And we investigate the relationship between them. Several numerical examples are shown to confirm the efficiency and reliability of the proposed methods

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“…Recently, the proposed elements are used to solve a driven cavity problem [17] and an interface problem governed by the Stokes, Darcy, and Brinkman equations [16].…”

Section: In §5mentioning

confidence: 99%

Stable cheapest nonconforming finite elements for the Stokes equations

Kim

Yim

Sheen

2016

Journal of Computational and Applied Mathematics

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“…The

{P}_1

–nonconforming quadrilateral finite element [27] has an advantage in computing stiffness matrice as the gradient of linear polynomials is constant in each quadrilateral as well as it has the smallest DOFs (degrees of freedom) for a given quadrilateral mesh. This finite element have been applied to fluid dynamics, elasticity, and electromagnetics [8, 15, 16, 23‐26, 28]. Unlikely other finite elements, the

{P}_1

–nonconforming finite element space is strongly tied with boundary conditions due to the element‐by‐element “dice rule constraint” (See (3.1)).…”

Section: Introductionmentioning

confidence: 99%

P1$$ {P}_1 $$–Nonconforming quadrilateral finite element space with periodic boundary conditions: Part I. Fundamental results on dimensions, bases, solvers, and error analysis

Yim

Sheen

2023

Numerical Methods Partial

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The P1$$ {P}_1 $$–nonconforming quadrilateral finite element space with periodic boundary conditions is investigated. The dimension and basis for the space are characterized by using the concept of minimally essential discrete boundary conditions. We show that the situation is different based on the parity of the number of discretizations on coordinates. Based on the analysis on the space, we propose several numerical schemes for elliptic problems with periodic boundary conditions. Some of these numerical schemes are related to solving linear equations consisting of non‐invertible matrices. By courtesy of the Drazin inverse, the existence of corresponding numerical solutions is guaranteed. The theoretical relation between the numerical solutions is derived, and it is confirmed by numerical results. Finally, the extension to the three dimensions is provided.

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“…The P 1 -nonconforming quadrilateral finite element [27] has an advantage in computing stiffness matrice as the gradient of linear polynomials is constant in each quadrilateral as well as it has the smallest number of DOFs (degrees of freedom) for given quadrilateral mesh. There have been a number of studies about this finite element for fluid dynamics, elasticity, electromagnetics [23,15,25,24,26,28,8,16]. Unlikely other finite elements, this space is strongly tied with the boundary condition for given problem due to the dice rule constraint element by element (See (3.1)).…”

mentioning

confidence: 99%

$P_1$--Nonconforming Quadrilateral Finite Element Space with Periodic Boundary Conditions: Part I. Fundamental results on dimensions, bases, solvers, and error analysis

Yim¹,

Sheen²

2022

Preprint

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The P 1 -nonconforming quadrilateral finite element space with periodic boundary condition is investigated. The dimension and basis for the space are characterized with the concept of minimally essential discrete boundary conditions. We show that the situation is totally different based on the parity of the number of discretization on coordinates. Based on the analysis on the space, we propose several numerical schemes for elliptic problems with periodic boundary condition. Some of these numerical schemes are related with solving a linear equation consisting of a non-invertible matrix. By courtesy of the Drazin inverse, the existence of corresponding numerical solutions is guaranteed. The theoretical relation between the numerical solutions is derived, and it is confirmed by numerical results. Finally, the extension to the three dimensional is provided.

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Nonconforming Finite Element Method Applied to the Driven Cavity Problem

Cited by 6 publications

References 35 publications

Stable cheapest nonconforming finite elements for the Stokes equations

Stable cheapest nonconforming finite elements for the Stokes equations

P1$$ {P}_1 $$–Nonconforming quadrilateral finite element space with periodic boundary conditions: Part I. Fundamental results on dimensions, bases, solvers, and error analysis

$P_1$--Nonconforming Quadrilateral Finite Element Space with Periodic Boundary Conditions: Part I. Fundamental results on dimensions, bases, solvers, and error analysis

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