What is the smallest possible constant in Céa’s lemma? (English) Zbl 1164.65495

Chen, Wei; Křížek, Michal

Appl. Math., Praha 51, No. 2, 129-144 (2006).

Summary: We consider finite element approximations of a second order elliptic problem on a bounded polytopic domain in \(\mathbb R^d\) with \(d\in \{1,2,3,\ldots \}\). The constant \(C\geq 1\) 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.

MSC:

65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15	Error bounds for boundary value problems involving PDEs
35J25	Boundary value problems for second-order elliptic equations

Keywords:

supercloseness; Lagrange finite elements; Lagrange remainder; lower estimates; second order elliptic problems; \(d\)-simplex; uniform partitions; linear simplicial elements; numerical results; interpolation error

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