Convergence of the nonconforming Wilson element for a class of nonlinear parabolic problems

Chou, S. H.; Li, Q.

doi:10.1090/S0025-5718-1990-1011439-X

Convergence of the nonconforming Wilson element for a class of nonlinear parabolic problems

Authors: S. H. Chou and Q. Li
Journal: Math. Comp. 54 (1990), 509-524
MSC: Primary 65N30
DOI: https://doi.org/10.1090/S0025-5718-1990-1011439-X
MathSciNet review: 1011439
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Abstract: This paper deals with the convergence properties of the nonconforming quadrilateral Wilson element for a class of nonlinear parabolic problems in two space dimensions. Optimal ${H^1}$ and ${L_2}$ error estimates for the continuous time Galerkin approximations are derived. It is also shown for rectangular meshes that the gradient of the Wilson element solution possesses superconvergence, and that the ${L_\infty }$ error on the gradient is of order $h\log (1/h)$.

Jim Douglas Jr. and Todd Dupont, Galerkin methods for parabolic equations, SIAM J. Numer. Anal. 7 (1970), 575–626. MR 277126, DOI https://doi.org/10.1137/0707048
P. Lesaint and M. Zlámal, Convergence of the nonconforming Wilson element for arbitrary quadrilateral meshes, Numer. Math. 36 (1980/81), no. 1, 33–52. MR 595805, DOI https://doi.org/10.1007/BF01395987
Zhong Ci Shi, A convergence condition for the quadrilateral Wilson element, Numer. Math. 44 (1984), no. 3, 349–361. MR 757491, DOI https://doi.org/10.1007/BF01405567
Shu Min Shen, $L_\infty $-convergence of nonconforming finite element approximations, Math. Numer. Sinica 8 (1986), no. 3, 225–230 (Chinese, with English summary). MR 868028

H. Wang, Superconvergence of the Wilson element, J. Shandong Univ. 21 (1986), 89-95.

Mary Fanett Wheeler, A priori $L_{2}$ error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J. Numer. Anal. 10 (1973), 723–759. MR 351124, DOI https://doi.org/10.1137/0710062