Théophile Chaumont-Frelet, Inria project-team Atlantis

We consider high-frequency wave propagation problems modeled by the Helmholtz equation and discretized by Lagrange finite elements. We derive novel guaranteed a posteriori error estimates. By guaranteed, we mean that the right-hand side of the error estimate is fully computable, and in particular, do not contain any "generic constant". The proposed error estimates are based on an a posteriori error estimation technique called "flux equilibration".

In the low-frequency regime, the problem becomes coercive, and it is well-known that flux equilibration provides guaranteed upper bound on the discretization error. The focus of this work is thus the high-frequency regime, where the problem lacks coercivity. We show that compared to the coercive case, an additional "pre-factor" term has to be added in the error estimate. This pre-factor increases with the frequency, but tends to zero as the mesh is refined. We propose several approaches to bound this pre-factor in a guaranteed manner, that we illustrate with numerical experiments.