Galerkin error reduction and eigenvalue localisation for conforming FEM

Abstract The reduction factor $$\varrho $$ ϱ in a Galerkin scheme with conforming $$P_1$$ P 1 finite element spaces on simplicial meshes is the maximal quotient of the Galerkin errors in the $$L^2$$ L 2 norm divided by the energy norm (the $$H^1$$ H 1 semi-norm) for the Poisson model problem for instance. An old and a new scheme allow for the computation of upper bounds of $$\varrho $$ ϱ on a given mesh. The schemes are compared theoretically and numerically before three applications for the guaranteed lower eigenvalue bounds (GLB) illustrate their impact. The three eigenvalue problems are named after Laplace, Schrödinger, and Steklov. The resulting GLB compete with those from recent post-processing of nonconforming Crouzeix–Raviart eigenvalue approximations in the classical and in extended versions.