Numer. Math. (2018) 138:681–721
Estimating and localizing the algebraic and total
numerical errors using ﬂux reconstructions
· Z. Strakoš
· M. Vohralík
Received: 4 May 2016 / Revised: 28 March 2017 / Published online: 6 September 2017
© Springer-Verlag GmbH Deutschland 2017
Abstract This paper presents a methodology for computing upper and lower bounds
for both the algebraic and total errors in the context of the conforming ﬁnite element
discretization of the Poisson model problem and an arbitrary iterative algebraic solver.
The derived bounds do not contain any unspeciﬁed constants and allow estimating the
local distribution of both errors over the computational domain. Combining these
bounds, we also obtain guaranteed upper and lower bounds on the discretization error.
This allows to propose novel mathematically justiﬁed stopping criteria for iterative
algebraic solvers ensuring that the algebraic error will lie below the discretization one.
Our upper algebraic and total error bounds are based on locally reconstructed ﬂuxes
in H(div,Ω), whereas the lower algebraic and total error bounds rely on locally con-
This work was supported by the ERC-CZ Project LL1202 ﬁnanced by the MŠMT of the Czech Republic,
and by the Project 13-06684S of the Grant Agency of the Czech Republic. It has also received funding
from the European Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation program (Grant agreement No. 647134 GATIPOR).
Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Prague, Czech
Institute of Computer Science, Czech Academy of Sciences, Pod Vodárenskou vˇeží 2, 182 07
Prague, Czech Republic
Inria Paris, 2 rue Simone Iff, 75589 Paris, France
Université Paris-Est, CERMICS (ENPC), 77455 Marne-la-Vallée 2, France