A posteriori error estimates via equilibrated fluxes 2015/2016
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: 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
    1. adaptation of spatial meshes
    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

Standard lectures Prague (Karlín)

Wednesday March 2, 2016, 14:00–15:30, lecture room KNM, 4th floor
Wednesday March 2, 2016, 15:40–17:10, lecture room KNM, 4th floor
Tuesday March 8, 2016, 15:00–16:30, lecture room KNM, 4th floor
Tuesday March 8, 2016, 16:40–18:10, lecture room KNM, 4th floor
Wednesday March 9, 2016, 14:00–15:30, lecture room KNM, 4th floor
Wednesday March 9, 2016, 15:40–17:10, lecture room KNM, 4th floor
Friday March 11, 2016, 10:40–12:10, lecture room KNM, 4th floor
Friday March 11, 2016, 12:20–13:50, lecture room KNM, 4th floor
Tuesday March 22, 2016, 15:00–16:30, lecture room KNM, 4th floor
Tuesday March 22, 2016, 16:40–18:10, lecture room KNM, 4th floor


Overview lectures Prague (Karlín)

Monday March 7, 2016, 9:00–10:30, lecture room K6
Monday March 21, 2016, 9:00–10:30, lecture room K6

Lecture notes

version of February 21, 2015 PDF icon
Examples of (written) examinations

2011 PDF icon
2012 PDF icon

Examination Prague



Books on the subject

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 PDF icon.
  2. 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 PDF icon.
  3. Hannukainen A., Stenberg R., Vohralík M. A unified framework for a posteriori error estimation for the Stokes problem. Numer. Math. 122 (2012), 725–769, journal version, preprint PDF icon .
  4. Ern A., Vohralík M. A posteriori error estimation based on potential and flux reconstruction for the heat equation. SIAM J. Numer. Anal. 48 (2010), 198–223, journal version, paper PDF icon .