A posteriori error estimates via equilibrated fluxes 2023/2024
Martin Vohralík

Course description
Numerical simulations have become a basic tool for approximation of various phenomena described by partial differential equations in the sciences, engineering, medicine, environment, and many other domains.

Questions of primordial interest are:
• How large is the overall error between the unkown exact solution and the available numerical approximation?
• Where does it come from? Insufficient spatial mesh resolution? Time stepping? Iterative linearization? Iterative algebraic solver? And where is it localized? In which part of the computational domain? On which time step?
• How to make the numerical simulation efficient? I.e., how to obtain the best possible result for the smallest possible price (computing time, memory usage)?
The theory of a posteriori error estimation and adaptivity allows to indicate answers to these questions. This course presents its basic principles for a selection of model problems. An abstract unified framework is derived, with applications to textbook numerical algorithms: spatial discretizations (finite element, finite volume, mixed finite element, discontinuous Galerkin), time steppings (higher-order), iterative linearizations (fixed point, Zarantonello, Newton), and iterative algebraic solvers (multigrid).
Course topics
1. Basic notions of an a posteriori estimate:
1. guaranteed upper and lower bounds
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 $H^1$ and $\boldsymbol{H}(\mathrm{div})$
3. primal and dual variational formulations, energy and complementary energy
4. Green theorem
5. Prager and Synge theorem
6. Poincaré–Friedrichs inequalities
7. residual of a partial differential equation
8. energy norm and dual norms
3. Model problems
1. the Laplace equation
2. the stationary linear advection–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 $H^1$ (conforming finite element spaces) and $\boldsymbol{H}(\mathrm{div})$ (Raviart–Thomas spaces) on simplicial meshes
2. local postprocessing
3. equilibration
5. A posteriori error estimates of the spatial discretization error
1. finite element method
2. finite volume method
3. mixed finite element method
4. discontinuous Galerkin method
6. A posteriori error estimates of the algebraic and linearization errors
1. arbitrary iterative algebraic solver/multigrid
2. arbitrary iterative linearization/fixed point, Zarantonello, Newton
7. Use of the estimates
2. adaptation of the time step
3. stopping criteria for linear solvers
4. stopping criteria for nonlinear solvers

Lectures

The lectures will be organized in a condensed way during the two weeks from April 8 to April 19, 2024 at MFF UK, Sokolovská 83, Karlín, Prague.
Schedule:

• Monday April 8, 2024, 12:20–13:50, lecture room K9
• Monday April 8, 2024, 14:00–15:30, lecture room K9
• Wednesday April 10, 2024, 10:40–12:10, lecture room K433
• Wednesday April 10, 2024, 12:20–13:50, lecture room K11
• Thursday April 11, 2024, 10:40–12:10, lecture room K358MUUK
• Thursday April 11, 2024, 12:20–13:50, lecture room K4
• Wednesday April 17, 2024, 10:40–12:10, lecture room K433
• Wednesday April 17, 2024, 12:20–13:50, lecture room K11
• Thursday April 18, 2024, 10:40–12:10, lecture room K358MUUK
• Thursday April 18, 2024, 12:20–13:50, lecture room K4
The lecture rooms will be set soon.
Presentations

A priori estimates, a posteriori estimates, $hp$ finite elements

Lecture notes

version of April 8, 2024
Examples of (written) examinations

2015
2018

Examination

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 .
5. Miraçi, A., Papež, J., and Vohralík, M. A-posteriori-steered p-robust multigrid with optimal step-sizes and adaptive number of smoothing steps. SIAM J. Sci. Comput. 43 (2021), S117–S145, Journal version, paper .