Asymptotic error expansion and richardson extrapolation for linear fine elements
Numerische Mathematik
On a global superconvergence of the gradient of linear triangular elements
Journal of Computational and Applied Mathematics
Superconvergence of the gradient of Galerkin approximations for elliptic problems
Journal of Computational and Applied Mathematics
SIAM Journal on Scientific and Statistical Computing
Ultraconvergence of the patch recovery technique II
Mathematics of Computation
Local and parallel finite element algorithms based on two-grid discretizations
Mathematics of Computation
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In this paper, we report some new ultraconvergence results of biquadratic Lagrange elements for the Dirichlet problem of Poisson's equation, -Δu = f. The point-line-area interpolant in [V. Girault, P.A. Raviart, A Finite Element Methods for Navier-Stokes Equation, Theory and Algorithms, Springer, 1986] is chosen in this paper, instead of the traditional pure point interpolant in [P.G. Ciarlet, Basic error estimates for elliptic problems, in: P.G. Ciarlet, J.L. Lions (Eds.), Finite Element Methods, Part 1, North-Holland, Amsterdam, 1991, pp. 17-351]. Suppose that the solution is smooth enough, by means of an a posteriori interpolant, the ultraconvergence O(h4) in H1 norm is proved for uniform rectangles □ij, and the higher ultraconvergence O(h6-l) in Hl(l = 0, 1) norm under the special case of uniform squares □ij and fxxyy = 0. Even when fxxyy ≠ 0, we propose two techniques: (1) the Richardson extrapolation method and (2) the correction method, to retain the same higher ultraconvergence results. Moreover, the ultraconvergence O(h6-l|ln h|) is also proved for l (l = 0, 1) order infinite norms. In this paper, the numerical experiments are provided to validate all the ultraconvergence results made. Note that the new ultraconvergence results under the special case are three order higher than the optimal convergence rate in [P.G. Ciarlet, Basic error estimates for elliptic problems, in: P.G. Ciarlet, J.L. Lions (Eds.), Finite Element Methods, Part 1, North-Holland, Amsterdam, 1991, pp. 17-351], and one order than that in [Q. Lin, N. Yah, A. Zhou, A rectangle test for interpolated finite elements, in: Proc. Sys. Sci. and Sys. Engrg., Great Wall Culture Publishers, Hong Kong, 1991, pp. 217-229].