A posteriori error estimates via equilibrated fluxes 2019/2020
Martin Vohralík

Course description
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.
Course topics
1. Basic notions of an a posteriori estimate:
1. guaranteed upper bound
2. local efficiency
3. asymptotic exactness
4. robustness with respect to parameters
5. evaluation cost
2. Fundamental physical and mathematical principles and theorems
1. constitutive law, equilibrium equation, constraint
2. continuity of potential and continuity of the normal trace of flux: the spaces H1 and H(div)
3. primal and dual variational formulations, energy and complementary energy
4. Green theorem
5. Prager and Synge theorem
6. Poincaré–Friedrichs–Wirtinger inequalities
7. residual of a partial differential equation
8. energy norm and dual norms
3. A unified framework for a posteriori error estimates
1. the Laplace equation
2. the stationary linear convection–diffusion–reaction equation
3. the Stokes problem
4. the heat equation
5. the nonlinear Laplace equation
4. Construction and evaluation of the estimators
1. approximation of the spaces H1 (conforming finite element spaces) and H(div) (Raviart–Thomas spaces) on simplicial meshes
2. local postprocessing
3. equilibration
5. Efficiency of the "residual" estimates
1. bubble functions
2. equivalence of norms on finite-dimensional spaces
3. inverse inequalities
6. Use of the estimates
2. adaptation of the time step
3. stopping criteria for linear solvers
4. stopping criteria for nonlinear solvers
7. Application to different numerical methods
1. finite element method
2. finite volume method
3. mixed finite element method
4. 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

version of April 16, 2018 Examples of (written) examinations

2011 2012 Examination Prague

Books on the subject
• 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.

Detailed material
1. 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 .
2. 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, journal version, preprint .
3. 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 .
4. 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 .