Abstract
We consider two-dimensional singularly perturbed fourth order problems and estimate on properly constructed layer-adapted errors of a mixed method in the associated energy norms and balanced norms. This paper is a shortened version of [4].
Highlights
Let us consider the singularly perturbed plate bending problem given by the fourth-order differential equation ε2∆2u − b∆u + (c · ∇)u + du = f in Ω := (0, 1)2, (1a)
The solution of this problem lies in H02 which means, a conforming finite element discretisation requires C1-elements
For our numerical analysis to work we assume a decomposition of the solution u of problem (1a)+(1b) into a smooth part, boundary layers and corner layers: u = S + Ek, where I = {1, 2, 3, 4, 12, 23, 34, 41}
Summary
The solution of this problem lies in H02 which means, a conforming finite element discretisation requires C1-elements. They are not very popular in 2d or 3d, which leads to the widely usage of mixed or non-conforming methods. For non-singularly perturbed problems (ε = 1) and p-th order finite-element approximation for u and w = ∆u, the classical error estimate u − uh 1 + h w − wh 0 ≤ Chp u p+1. Perturbed fourth order problems for the discrete solutions uh and wh on a standard shape-regular mesh with p ≥ 2, is known, see [3, 11].
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