A posteriori error estimates via equilibrated fluxes 2021/2022 Martin Vohralík
Numerical simulations have become a basic tool for approximation of various phenomena in the sciences, engineering, medicine, and many other domains.
Two questions of primordial interest are:
How large is the overall error between the exact and approximate solutions and where is it localized?
How to make the numerical simulation efficient - obtain as good as possible result for as small as possible price (calculation time, memory usage)?
The theory of a posteriori error estimation allows to give/indicate answers to these questions. This course presents its basic principles for model problems. An abstract unified framework is derived. Applications to the finite element, finite volume, mixed finite element, and discontinuous Galerkin methods are given.
Basic notions of an a posteriori estimate:
guaranteed upper bound
robustness with respect to parameters
Fundamental physical and mathematical principles and theorems
Ainsworth, M., Oden, J.T., A posteriori error estimation in finite element analysis.
Wiley-Interscience, New York, 2000.
Repin, S.I., A posteriori estimates for partial differential equations.
Walter de Gruyter GmbH & Co. KG, Berlin, 2008.
Verfürth, R., A posteriori error estimation techniques for finite element methods.
Oxford University Press, Oxford, 2013.
Ern A., Vohralík, M. Polynomial-degree-robust a posteriori estimates in a unified setting for
conforming, nonconforming, discontinuous Galerkin, and mixed discretizations. SIAM J. Numer. Anal.53 (2015), 1058–1081,
Journal version, paper .
Čermák M., Hecht F., Tang Z., Vohralík M. Adaptive inexact iterative algorithms based on
polynomial-degree-robust a posteriori estimates for the Stokes problem. Numer. Math.138 (2018), 1027–1065,
Ern A., Smears I., Vohralík M. Guaranteed, locally space-time efficient, and polynomial-degree robust
a posteriori error estimates for high-order discretizations of parabolic problems. SIAM J. Numer. Anal.55 (2017), 2811–2834,
journal version, paper
Ern A., Vohralík M. Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear
diffusion PDEs. SIAM J. Sci. Comput.35 (2013), A1761–A1791, Journal version, paper .