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: 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
    1. adaptation of spatial meshes
    2. adaptation of the time step
    3. stopping criteria for linear solvers
    4. stopping criteria for nonlinear solvers


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.

The lecture rooms will be set soon.

A priori estimates, a posteriori estimates, \(hp\) finite elements PDF icon
A posteriori estimates (heat, advection–reaction–diffusion, and nonlinear Laplace equations) and adaptivity PDF icon

Lecture notes

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

2015 PDF icon
2018 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.
  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 PDF icon.