A posteriori error estimates via equilibrated fluxes 2019/2020 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
continuity of potential and continuity of the normal trace of flux: the spaces H1 and H(div)
primal and dual variational formulations, energy and complementary energy
Prager and Synge theorem
residual of a partial differential equation
energy norm and dual norms
A unified framework for a posteriori error estimates
the Laplace equation
the stationary linear convection–diffusion–reaction equation
the Stokes problem
the heat equation
the nonlinear Laplace equation
Construction and evaluation of the estimators
approximation of the spaces H1 (conforming finite element spaces) and H(div) (Raviart–Thomas spaces) on simplicial meshes
Efficiency of the "residual" estimates
equivalence of norms on finite-dimensional spaces
Use of the estimates
adaptation of spatial meshes
adaptation of the time step
stopping criteria for linear solvers
stopping criteria for nonlinear solvers
Application to different numerical methods
finite element method
finite volume method
mixed finite element method
discontinuous Galerkin method
Lectures: April 6 – April 22 2020 (MFF UK, Sokolovská 83, Karlín, Prague)
Monday April 6, 2020, 10:40–12:10, lecture room K9, ground floor Monday April 6, 2020, 12:20–13:50, lecture room K5, ground floor The lectures in person are cancelled because of the coronavirus outbreak and replaced by online lecturing after mutual arrangement. Lecture notes
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 .