Volume 47, Number 3, May-June 2013
|Page(s)||689 - 715|
|Published online||04 March 2013|
A piecewise P2-nonconforming quadrilateral finite element
1 Department of Mathematics, Seoul
National University, 151-747
2 School of Mathematical Sciences, Dalian University of Technology, Dalian, China
3 Interdisciplinary Program in Computational Sciences and Technology, Seoul National University, 151-747 Seoul, Korea
4 Department of Mathematics, Konkuk University, 143-701 Seoul, Korea
Revised: 16 July 2012
We introduce a piecewise P2-nonconforming quadrilateral finite element. First, we decompose a convex quadrilateral into the union of four triangles divided by its diagonals. Then the finite element space is defined by the set of all piecewise P2-polynomials that are quadratic in each triangle and continuously differentiable on the quadrilateral. The degrees of freedom (DOFs) are defined by the eight values at the two Gauss points on each of the four edges plus the value at the intersection of the diagonals. Due to the existence of one linear relation among the above DOFs, it turns out the DOFs are eight. Global basis functions are defined in three types: vertex-wise, edge-wise, and element-wise types. The corresponding dimensions are counted for both Dirichlet and Neumann types of elliptic problems. For second-order elliptic problems and the Stokes problem, the local and global interpolation operators are defined. Also error estimates of optimal order are given in both broken energy and L2(Ω) norms. The proposed element is also suitable to solve Stokes equations. The element is applied to approximate each component of velocity fields while the discontinuous P1-nonconforming quadrilateral element is adopted to approximate the pressure. An optimal error estimate in energy norm is derived. Numerical results are shown to confirm the optimality of the presented piecewise P2-nonconforming element on quadrilaterals.
Mathematics Subject Classification: 65N30 / 76M10
Key words: Nonconforming finite element / Stokes problem / elliptic problem / quadrilateral
© EDP Sciences, SMAI, 2013
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