Locally stabilized P1-nonconforming quadrilateral and hexahedral finite element methods for the Stokes equations

Authors:
Xinlong Feng;Imbunm Kim;Hyun Nam;Dongwoo Sheen
Affiliations:
Department of Mathematics, Seoul National University, Seoul 151-747, Republic of Korea and College of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, PR China;Department of Mathematics, Seoul National University, Seoul 151-747, Republic of Korea;Interdisciplinary Program in Computational Science & Technology, Seoul National University, Seoul 151-747, Republic of Korea;Department of Mathematics, Seoul National University, Seoul 151-747, Republic of Korea and Interdisciplinary Program in Computational Science & Technology, Seoul National University, Seoul 151-747 ...
Venue:
Journal of Computational and Applied Mathematics
Year:
2011

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Abstract

In this paper, we consider locally stabilized pairs (P"1,P"1)-nonconforming quadrilateral and hexahedral finite element methods for the two- and three-dimensional Stokes equations. The stabilization is obtained by adding to the bilinear form the difference between an exact Gaussian quadrature rule for quadratic polynomials and an exact Gaussian quadrature rule for linear polynomials. Optimal error estimates are derived in the energy norm and the L^2-norm for the velocity and in the L^2-norm for the pressure. In addition, numerical experiments to confirm the theoretical results are presented. From our numerical results, we observe that the proposed stabilized(P"1,P"1)-nonconforming finite element method shows better performance than the standard method.