Optimal meshes for finite elements of arbitrary order. (English) Zbl 1202.65015

Given a function \(f\) on a bounded domain \(\Omega \subset \mathbb R^2,\) the interpolation by finite elements with piecewise polynomials of degree \(m-1\) is studied. Specifically, there is the interest in minimizing the \(L_p\) error by optimizing the mesh such that a (nearly) minimal \(L_p\) norm of the interpolation error is achieved. To this, end the interpolation of homogeneous polynomials of degree \(m\) is studied. The results refer to the interpolation problem, while the optimization of the meshes for the solution of elliptic differential equations is governed by a different analysis.


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
65N15 Error bounds for boundary value problems involving PDEs
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