New error estimates of biquadratic Lagrange elements for Poisson's equation

Abstract

In this paper, we report some new ultraconvergence results of biquadratic Lagrange elements for the Dirichlet problem of Poisson's equation, − Δ u = 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 ( h 4 ) in H 1 norm is proved for uniform rectangles □ i j , and the higher ultraconvergence O ( h 6 − ℓ ) in H ℓ ( ℓ = 0 , 1 ) norm under the special case of uniform squares □ i j and f x x y y = 0 . Even when f x x y y ≠ 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 ( h 6 − 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. Yan, 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].