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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 \(\Delta^2u=f\) with the homogeneous Dirichlet conditions on the boundary of \(\bar\Omega\) – 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 \(u\in H^4(\Omega)\) but without such widely used restriction as a quasiuniform structure of the grid. The authors give an estimate of the type \(O(h_{max}^2)\) in a grid norm. They also present numerical examples for the grids of type \(128\times 128\) and \(64\times 384\).
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 for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35J40 Boundary value problems for higher-order elliptic equations
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