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Anisotropic interpolation with applications to the finite element method. (English) Zbl 0746.65077

For the approximation of anisotropic structures such as edges, boundary or interior layers, it is natural to use a finite element mesh with different mesh sizes in different directions. The authors extend the usual Bramble-Hilbert theory for proving more refined estimates of the interpolation error. These results are applied to common finite elements, and it is observed that elements with a large angle may be useful for approximating anisotropic structures.

The difference between the interpolation and the finite element approximation error is discussed and some results for rectangular elements are derived. The results are applied to the finite element approximation of elliptic equations on domains with edges.

The authors also suggest that it may be interesting to extend their results for other approximation operators [see P. Clément, Revue Franc. Automat. Inform. Rech. Opérat. 9, R-2, 77-84 (1975; Zbl 0368.65008); L. R. Scott and S. Zhang, Math. Comput. 54, No. 190, 483-493 (1990; Zbl 0696.65007)].

MSC:
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
41A05Interpolation (approximations and expansions)
65N15Error bounds (BVP of PDE)
41A63Multidimensional approximation problems
35J25Second order elliptic equations, boundary value problems
References:
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