Vassilevski, Yu. V.; Agouzal, A. Unified asymptotic analysis of interpolation errors for optimal meshes. (English. Russian original) Zbl 1156.65321 Dokl. Math. 72, No. 3, 879-882 (2005); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 405, No. 3, 295-298 (2005). Summary: The progress in adaptive mesh generation achieved over recent years makes it possible to attempt to generate optimal meshes or, at least, their approximations. For problems with anisotropic solutions, optimal meshes are also anisotropic. Therefore, estimates of interpolation errors for anisotropic meshes are required for approximation analysis of optimal meshes. Such error estimates in \(L_\infty\) for optimal triangulations were derived by Yu. V. Vassilevskii and K. N. Lipnikov [Comput. Math. Math. Phys. 39, No.9, 1468–1486 (1999); translation from Zh. Vychisl. Mat. Mat. Fiz. 39, No. 9, 1532–1551 (1999; Zbl 0981.65141)] and by A. Agouzal, K. Kipnikov, and Yu. Vassilevski [East-West J. Numer. Math. 7, No. 4, 223–244 (1999; Zbl 0946.65125)], for the two- and three-dimensional cases, respectively. However, the proofs in are not analogous and contain some inaccuracies concerning the lower bound on the interpolation error for an optimal mesh. More specifically, both proofs are I somewhat incomplete in the case of indefinite Hessians on arbitrary simplices. Since optimal meshes can have arbitrary elements, these proofs have to be refined. Moreover, the estimates for the interpolation error in \(L_\infty\) can easily be extended to \(L_p\). We present a complete corrected proof of the error estimate that unifies the two- and three-dimensional cases and, then, extend the result to estimates in \(L_p\). Cited in 3 Documents MSC: 65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations 65N15 Error bounds for boundary value problems involving PDEs Keywords:adaptive mesh generation; interpolation errors; error estimates; optimal meshes Citations:Zbl 0981.65141; Zbl 0946.65125 PDFBibTeX XMLCite \textit{Yu. V. Vassilevski} and \textit{A. Agouzal}, Dokl. Math. 72, No. 3, 879--882 (2005; Zbl 1156.65321); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 405, No. 3, 295--298 (2005)