# zbMATH — the first resource for mathematics

Optimal triangulations: existence, approximations, and double differentiation of $$P_1$$ finite element functions. (Russian, English) Zbl 1076.65112
Zh. Vychisl. Mat. Mat. Fiz. 43, No. 6, 866-874 (2003); translation in Comput. Math. Math. Phys. 43, No. 6, 827-835 (2003).
The authors introduce a concept of optimal unstructured triangulation and prove its existence under certain assumptions. Estimations of the $$L_{\infty}$$-error are presented for the operator of piecewise-linear interpolation. Quasi-optimal triangulations are found, which are approximations of optimal ones. The methodology relies upon the recovery of the Hessian of the piecewise smooth continuous solutions.

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