A posteriori error estimates via equilibrated fluxes 2021/2022
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


The lectures will be organized in a condensed way during the two weeks from April 25 to May 6, 2022 at MFF UK, Sokolovská 83, Karlín, Prague.

Lecture notes

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

2011 PDF icon
2012 PDF icon


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. 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 PDF icon .
  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 PDF icon .
  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 PDF icon.