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On the anisotropic accuracy analysis of ACM’s nonconforming finite element. (English) Zbl 1087.65103

The paper deals with the accuracy of a nonconforming finite element methods for the well-known equation ${{\Delta }}^{2}u=f$ with the homogeneous Dirichlet conditions on the boundary of $\overline{{\Omega }}$ – a union of several rectangles.

On nonquasiuniform rectangular grids, the so called ACM’s nonconforming finite elements are considered. The main aim is to study the convergence under the usual assumption that $u\in {H}^{4}\left({\Omega }\right)$ but without such widely used restriction as a quasiuniform structure of the grid. The authors give an estimate of the type $O\left({h}_{max}^{2}\right)$ in a grid norm. They also present numerical examples for the grids of type $128×128$ and $64×384$.

In places the presentation is rather strange – for example, the authors call the equation “biharmonic” and write that “the domain is the union of rectangles”.

##### MSC:
 65N15 Error bounds (BVP of PDE) 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE) 65N12 Stability and convergence of numerical methods (BVP of PDE) 35J40 Higher order elliptic equations, boundary value problems