Supercloseness of orthogonal projections onto nearby finite element spaces

Gawlik, Evan S.; Lew, Adrian J.

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 2, pp. 559-576.

We derive upper bounds on the difference between the orthogonal projections of a smooth function $u$ onto two finite element spaces that are nearby, in the sense that the support of every shape function belonging to one but not both of the spaces is contained in a common region whose measure tends to zero under mesh refinement. The bounds apply, in particular, to the setting in which the two finite element spaces consist of continuous functions that are elementwise polynomials over shape-regular, quasi-uniform meshes that coincide except on a region of measure $O\left(h^{\gamma }\right)$, where $\gamma$ is a nonnegative scalar and $h$ is the mesh spacing. The projector may be, for example, the orthogonal projector with respect to the $L^2$- or $H^1$-inner product. In these and other circumstances, the bounds are superconvergent under a few mild regularity assumptions. That is, under mesh refinement, the two projections differ in norm by an amount that decays to zero at a faster rate than the amounts by which each projection differs from $u$. We present numerical examples to illustrate these superconvergent estimates and verify the necessity of the regularity assumptions on $u$.

A.B. Andreev, Supercloseness between the elliptic projection and the approximate eigenfunction and its application to a postprocessing of finite element eigenvalue problems. In Numer. Anal. Appl. Springer (2005) 100–107. | Zbl

I. Babuška and A. Miller, The post-processing approach in the finite element method, Part 1: Calculation of displacements, stresses and other higher derivatives of the displacements. Int. J. Numer. Methods Eng. 20 (1984) 1085–1109. | DOI | Zbl

I. Babuška, T. Strouboulis, C.S. Upadhyay and S.K. Gangaraj, Computer-based proof of the existence of superconvergence points in the finite element method; superconvergence of the derivatives in finite element solutions of Laplace’s, Poisson’s, and the elasticity equations. Numer. Methods Partial Differ. Equ. 12 (1996) 347–392. | DOI | MR | Zbl

R.E. Bank and J. Xu, Asymptotically exact a posteriori error estimators, Part I: Grids with superconvergence. SIAM J. Numer. Anal. 41 (2003) 2294–2312. | DOI | MR | Zbl

J. Barlow, Optimal stress locations in finite element models. Int. J. Numer. Methods Eng. 10 (1976) 243–251. | DOI | Zbl

J. Brandts and M. Křížek, Gradient superconvergence on uniform simplicial partitions of polytopes. IMA J. Numer. Anal. 23 (2003) 489–505. | DOI | MR | Zbl

B. Cockburn, M. Luskin, C.W. Shu and E. Süli, Enhanced accuracy by post-processing for finite element methods for hyperbolic equations. Math. Comput. 72 (2003) 577–606. | DOI | MR | Zbl

M. Crouzeix and V. Thomee, Stability in $L^p$ and $W^{1,p}$ of the $L^2$-projection onto finite element function spaces. Math. Comput. 48 (1987) 531–532. | MR | Zbl

A. Ern and J.L. Guermond, Theory and Practice of Finite Elements. Springer, New York (2004). | MR | Zbl

E.S. Gawlik and A.J. Lew, Unified analysis of finite element methods for problems with moving boundaries (2014). | MR

G. Goodsell, Pointwise superconvergence of the gradient for the linear tetrahedral element. Numer. Methods Partial Differ. Equ. 10 (1994) 651–666. | DOI | MR | Zbl

G. Goodsell and J.R. Whiteman, A unified treatment of superconvergent recovered gradient functions for piecewise linear finite element approximations. Int. J. Numer. Methods Eng. 27 (1989) 469–481. | DOI | MR | Zbl

Y. Huang and J. Xu, Superconvergence of quadratic finite elements on mildly structured grids. Math. Comput. 77 (2008) 1253–1268. | DOI | MR | Zbl

M. Křížek and P. Neittaanmäki, Superconvergence phenomenon in the finite element method arising from averaging gradients. Numer. Math. 45 (1984) 105–116. | DOI | MR | Zbl

M. Křížek and P. Neittaanmäki, On superconvergence techniques. Acta Appl. Math. 9 (1987) 175–198. | DOI | MR | Zbl

B. Li, Lagrange interpolation and finite element superconvergence. Numer. Methods Partial Differ. Equ. 20 (2004) 33–59. | DOI | MR | Zbl

J. Liu, G. Hu and Q. Zhu, Superconvergence of tetrahedral quadratic finite elements for a variable coefficient elliptic equation. Numer. Methods Partial Differ. Equ. 29 (2012) 1043–1055. | DOI | MR | Zbl

L.A. Oganesyan and L.A. Rukhovets, Study of the rate of convergence of variational difference schemes for second-order elliptic equations in a two-dimensional field with a smooth boundary. USSR Comput. Math. Math. Phys. 9 (1969) 158–183. | DOI | Zbl

A.H. Schatz, I.H. Sloan and L.B. Wahlbin, Superconvergence in finite element methods and meshes that are locally symmetric with respect to a point. SIAM J. Numer. Anal. 33 (1996) 505–521. | DOI | MR | Zbl

L.B. Wahlbin, Superconvergence in Galerkin finite element methods. Springer, Berlin (1995). | MR | Zbl

O.C. Zienkiewicz and J.Z. Zhu, The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique. Int. J. Numer. Methods Eng. 33 (1992) 1331–1364. | DOI | MR | Zbl