# Article

**Keywords:**

second-order boundary value problems; a posteriori error estimates; complementary energy; Friedrichs’ inequality; numerical example

**Summary:**

This contribution shows how to compute upper bounds of the optimal constant in Friedrichs' and similar inequalities. The approach is based on the method of $a priori-a posteriori inequalities$ [9]. However, this method requires trial and test functions with continuous second derivatives. We show how to avoid this requirement and how to compute the bounds on Friedrichs' constant using standard finite element methods. This approach is quite general and allows variable coefficients and mixed boundary conditions. We use the computed upper bound on Friedrichs' constant in a posteriori error estimation to obtain guaranteed error bounds.