Mathematikgebäude, Raum M 511

The construction of a finite element space usually consists of two steps. First, a broken space is obtained prescribing a finite element in each cell of a mesh. Second, certain adjacent degrees of freedom are identified, and the associated shape functions are coupled, so to enforce some sort of continuity. Hence, when approximating a target function, the best error in the resulting finite element space is not smaller than the one in the associated broken space. We prove that the converse inequality is true, up to a constant, if and only if a suitable algebraic condition holds. We show also that a quantitative control of the best such constant can be obtained by averaging operators. In certain cases, the analysis can be further refined, noticing that the best errors in the two spaces actually coincide. We illustrate our findings by several examples. This is a joint work with Thirupathi Gudi and Andreas Veeser.

Dr. Pietro Zanotti

Università degli Studi di Milano