Algebraic and discretization error estimation by equilibrated fluxes for discontinuous Galerkin methods on nonmatching grids

Vít Dolejší (1) , Ivana Šebestová (1) , Martin Vohralík (2)

1 Department of Numerical Mathematics
2 SERENA - Simulation for the Environment: Reliable and Efficient Numerical Algorithms

We derive a posteriori error estimates for the discontinuous Galerkin method applied to the Poisson equation. We allow for a variable polynomial degree and simplicial meshes with hanging nodes and propose an approach allowing for simple (nonconforming) flux reconstructions in such a setting. We take into account the algebraic error stemming from the inexact solution of the associated linear systems and propose local stopping criteria for iterative algebraic solvers. An algebraic error flux reconstruction is introduced in this respect. Guaranteed reliability and local efficiency are proven. We next propose an adaptive strategy combining both adaptive mesh refinement and adaptive stopping criteria. At last, we detail a form of the estimates avoiding any practical reconstruction of a flux and only working with the approximate solution, which simplifies greatly their evaluation. Numerical experiments illustrate a tight control of the overall error, good prediction of the distribution of both the discretization and algebraic error components, and efficiency of the adaptive strategy.

Mots clés: adaptivity; discontinuous Galerkin method; linear diffusion problems; hanging nodes; stopping criteria; a posteriori estimate; flux reconstruction; error components; simple evaluation; distribution of the error

Domaines: Analyse numérique [math.NA]

Vít Dolejší, Ivana Šebestová, Martin Vohralík. Algebraic and discretization error estimation by equilibrated fluxes for discontinuous Galerkin methods on nonmatching grids. Journal of Scientific Computing, 2015, 64 (1), pp.1-34. ⟨10.1007/s10915-014-9921-2⟩. ⟨hal-00851822v2⟩