We derive an optimal lower bound of the interpolation error for linear finite elements on a bounded two-dimensional domain. Using the supercloseness between the linear interpolant of the true solution of an elliptic problem and its finite element solution on uniform partitions, we further obtain two-sided a priori bounds of the discretization error by means of the interpolation error. Two-sided bounds for bilinear finite elements are given as well. Numerical tests illustrate our theoretical analysis.
Mots-clés : Lagrange finite elements, Céa's lemma, superconvergence, lower error estimates
@article{M2AN_2011__45_5_915_0, author = {K\v{r}{\'\i}\v{z}ek, Michal and Roos, Hans-Goerg and Chen, Wei}, title = {Two-sided bounds of the discretization error for finite elements}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {915--924}, publisher = {EDP-Sciences}, volume = {45}, number = {5}, year = {2011}, doi = {10.1051/m2an/2011003}, mrnumber = {2817550}, zbl = {1269.65113}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/m2an/2011003/} }
TY - JOUR AU - Křížek, Michal AU - Roos, Hans-Goerg AU - Chen, Wei TI - Two-sided bounds of the discretization error for finite elements JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2011 SP - 915 EP - 924 VL - 45 IS - 5 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2011003/ DO - 10.1051/m2an/2011003 LA - en ID - M2AN_2011__45_5_915_0 ER -
%0 Journal Article %A Křížek, Michal %A Roos, Hans-Goerg %A Chen, Wei %T Two-sided bounds of the discretization error for finite elements %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2011 %P 915-924 %V 45 %N 5 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2011003/ %R 10.1051/m2an/2011003 %G en %F M2AN_2011__45_5_915_0
Křížek, Michal; Roos, Hans-Goerg; Chen, Wei. Two-sided bounds of the discretization error for finite elements. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 5, pp. 915-924. doi : 10.1051/m2an/2011003. http://archive.numdam.org/articles/10.1051/m2an/2011003/
[1] Gradient superconvergence on uniform simplicial partitions of polytopes. IMA J. Numer. Anal. 23 (2003) 489-505. | MR | Zbl
and ,[2] What is the smallest possible constant in Céa's lemma? Appl. Math. 51 (2006) 128-144. | Zbl
and ,[3] Lower bounds for the interpolation error for finite elements. Mathematics in Practice and Theory 39 (2009) 159-164 (in Chinese). | MR | Zbl
and ,[4] The finite element method for elliptic problems. North-Holland, Amsterdam (1978). | MR | Zbl
,[5] Superconvergence analysis of the Galerkin FEM for a singularly perturbed convection-diffusion problems with characteristic layers. Numer. Methods Partial Differ. Equ. 24 (2008) 144-164. | MR | Zbl
and ,[6] Numerical treatment of partial differential equations. Springer-Verlag, Berlin, Heidelberg (2007). | MR | Zbl
, and ,[7] Two-sided a posteriori error estimates for linear elliptic problems with mixed boundary conditions. Appl. Math. 52 (2007) 235-249. | MR | Zbl
,[8] Finite element approximation of variational problems and applications. Longman Scientific & Technical, Harlow (1990). | Zbl
and ,[9] Mathematical and Numerical Modelling in Electrical Engineering: Theory and Applications. Kluwer, Dordrecht (1996). | MR | Zbl
and ,[10] Finite element methods: Accuracy and improvement. Science Press, Beijing (2006).
and ,[11] Introduction aux méthodes des éléments finis. Mir, Moscow (1985). | Zbl
and ,[12] Mathematical theory of elastic and elasto-plastic bodies: An introduction. Elsevier, Amsterdam (1981). | Zbl
and ,[13] An investigation of the rate of convergence of variational-difference schemes for second order elliptic equations in a two-dimensional region with smooth boundary. Ž. Vyčisl. Mat. i Mat. Fyz. 9 (1969) 1102-1120. | MR | Zbl
and ,[14] An analysis of the finite element method. Prentice-Hall, Inc., Englewood Cliffs, New Jersey (1973). | MR | Zbl
and ,[15] A review of a posteriori error estimation and adaptive mesh-refinement techniques. John Wiley & Sons, Chichester, Teubner, Stuttgart (1996). | Zbl
,[16] Superconvergence in Galerkin finite element methods, Lect. Notes in Math. 1605. Springer, Berlin (1995). | MR | Zbl
,[17] Analysis of recovery type a posteriori error estimation for mildly structured grids. Math. Comp. 73 (2004) 1139-1152. | MR | Zbl
and ,[18] Superconvergence analysis and a posteriori error estimation in finite element methods. Science Press, Beijing (2008).
,Cité par Sources :