Optimally adapted meshes for finite elements of arbitrary order and \(W^{1,p}\) norms. (English) Zbl 1238.65116

Given a function \(f\) defined on a bounded polygonal domain \(\Omega \subset {\mathbb{R}}^2\) and a number \(N>0\), the author studies the properties of the triangulation \({\mathcal{T}}_N\) that minimizes the distance between \(f\) and its interpolation on the associated finite element space, over all triangulations of at most \(N\) elements. The error is studied in the \(W^{1,p}\) semi-norm for \(1 \leq p < \infty\), and he consider Lagrange finite elements of arbitrary polynomial order \(m-1\). He establishes sharp asymptotic error estimates as \(N \rightarrow +\infty\) when the optimal anisotropic triangulation is used. A similar problem has been studied in other papers but with the error measured in the \(L^{p}\) norm. The extension of this analysis to the \(W^{1,p}\) norm is required in order to match more closely the needs of numerical analysis of partial differential equations, and it is not straightforward. In particular, the meshes which satisfy the optimal error estimate are characterized by a metric describing the local aspect ratio of each triangle and by a geometric constraint on their maximal angle, a second feature that does not appear for the \(L^{p}\) error norm. His analysis also provides with practical strategies for designing meshes such that the interpolation error satisfies the optimal estimate up to a fixed multiplicative constant.


65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65D05 Numerical interpolation
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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