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**Optimal finite-element interpolation on curved domains.**
*(English)*
Zbl 0678.65003

Interpolation of functions by the finite element method is studied. Interpolation results that are optimal with respect to the order of the finite elements and to the smoothness of the interpolated function are sought. Simplicial finite elements only are considered. The interpolated functions are not smooth (e.g. not continuous), and a generalized interpolation operator must be introduced via a local \(L^ 2\)- projection. The functions are defined on a domain with a curved piecewise smooth boundary.

A regular family of triangulations of the general curved domain made of curved d-simplices is associated with the corresponding finite element space. Exact triangulations of piecewise smooth domains are constructed. Conformity and interpolation with boundary conditions are discussed. Triangulation of the unit ball in \({\mathbb{R}}^ 3\) is built as an example of application.

A regular family of triangulations of the general curved domain made of curved d-simplices is associated with the corresponding finite element space. Exact triangulations of piecewise smooth domains are constructed. Conformity and interpolation with boundary conditions are discussed. Triangulation of the unit ball in \({\mathbb{R}}^ 3\) is built as an example of application.

Reviewer: V.Burjan

### MSC:

65D05 | Numerical interpolation |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |