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supercloseness; Lagrange finite elements; Lagrange remainder; lower estimates; elliptic problems; $d$-simplex; uniform partitions
We consider finite element approximations of a second order elliptic problem on a bounded polytopic domain in $\mathbb{R}^d$ with $d\in \lbrace 1,2,3,\ldots \rbrace $. The constant $C\ge 1$ appearing in Céa’s lemma and coming from its standard proof can be very large when the coefficients of an elliptic operator attain considerably different values. We restrict ourselves to regular families of uniform partitions and linear simplicial elements. Using a lower bound of the interpolation error and the supercloseness between the finite element solution and the Lagrange interpolant of the exact solution, we show that the ratio between discretization and interpolation errors is equal to $1+\mathcal O(h)$ as the discretization parameter $h$ tends to zero. Numerical results in one and two-dimensional case illustrating this phenomenon are presented.
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