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
Basic notions of an a posteriori estimate:
guaranteed upper and lower bounds
local efficiency
asymptotic exactness
robustness with respect to parameters
evaluation cost
Fundamental physical and mathematical principles and theorems
continuity of potential and continuity of the normal trace of flux: the spaces \(H^1\) and \(\boldsymbol{H}(\mathrm{div})\)
primal and dual variational formulations, energy and complementary energy
Green theorem
Prager and Synge theorem
Poincaré–Friedrichs inequalities
residual of a partial differential equation
energy norm and dual norms
Model problems
the Laplace equation
the stationary linear advection–diffusion–reaction equation
the Stokes problem
the heat equation
the nonlinear Laplace equation
Construction and evaluation of the estimators
approximation of the spaces \(H^1\) (conforming finite element spaces) and \(\boldsymbol{H}(\mathrm{div})\) (Raviart–Thomas spaces) on simplicial meshes
local postprocessing
equilibration
A posteriori error estimates of the spatial discretization error
finite element method
finite volume method
mixed finite element method
discontinuous Galerkin method
A posteriori error estimates of the algebraic and linearization errors
arbitrary iterative algebraic solver/multigrid
arbitrary iterative linearization/fixed point, Zarantonello, Newton
Use of the estimates
adaptation of spatial meshes
adaptation of the time step
stopping criteria for linear solvers
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
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
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 .
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 .
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
.
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 .
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 .