A Posteriori Numerical Analysis Based on the Method of Equilibrated Fluxes - NMNV464
Title: Aposteriorní numerická analýza metodou vyvážených toků
Guaranteed by: Department of Numerical Mathematics (32-KNM)
Faculty: Faculty of Mathematics and Physics
Actual: from 2023
Semester: summer
E-Credits: 3
Hours per week, examination: summer s.:2/0, Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: taught
Language: Czech, English
Teaching methods: full-time
Teaching methods: full-time
Guarantor: prof. Ing. Martin Vohralík, Ph.D.
Class: M Mgr. NVM
M Mgr. NVM > Povinně volitelné
Classification: Mathematics > Numerical Analysis
Incompatibility : NNUM054
Interchangeability : NNUM054
Is interchangeable with: NNUM054
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Annotation -
Last update: T_KNM (27.04.2015)
The course treats estimates of the error in approximate numerical solution of partial differential equations. The impact is on guaranteed and fully computable estimates. A unified framework for classical numerical methods (FEM, DGFEM,...) is introduced. The theory is developed for a large variety of problems. The emphasis is on the use of the estimates for efficient numerical calculation (adaptive mesh refinement, adaptive choice of the time step, stopping criteria for linear and nonlinear solvers).
Aim of the course -
Last update: T_KNM (02.04.2015)

The course derives fully computable estimates on the error in numerical solution of partial differential equations via the method of equilibrated fluxes.
Course completion requirements -
Last update: prof. Ing. Martin Vohralík, Ph.D. (11.06.2019)

Written examination
Literature - Czech
Last update: T_KNM (02.04.2015)

Vohralík, M., A posteriori error estimates for efficiency and error control in numerical simulations, skripta.

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.

Requirements to the exam -
Last update: prof. Ing. Martin Vohralík, Ph.D. (11.06.2019)

Students will be examined by written test with questions corresponding to the material addressed at the lectures.
Syllabus -
Last update: doc. RNDr. Václav Kučera, Ph.D. (19.12.2018)

Basic properties of an a posteriori estimate: guaranteed upper bound, local efficiency, asymptotic exactness, robustness with respect to parameters, low evaluation cost, distinction of error components.

Mathematical framework: continuity of the potential and continuity of the normal trace of the flux: the spaces H1 and H(div), primal and dual variational formulations, Green theorem, Prager and Synge theorem, Poincaré-Friedrichs-Wirtinger inequalities, residual of a partial differential equation, energy norm and dual norms.

Construction and evaluation of the estimators: potential reconstruction, flux reconstruction, equilibration using the mixed finite element method, equivalence with the error.

Theory for model problems: Laplace equation, the advection-diffusion-reaction equation, the Stokes equation, the unsteady heat equation, the nonlinear Laplace equation.

Application to classical numerical methods: conforming finite element method, nonconforming finite element method, mixed finite element method, discontinuous Galerkin method, finite volume method.

Use of the estimates: adaptation of spatial meshes, adaptation of the time step, stopping criteria for linear solvers, stopping criteria for nonlinear solvers.

Entry requirements -
Last update: prof. Ing. Martin Vohralík, Ph.D. (17.05.2019)

Theory of the linear elliptic partial differential equations of second

order, basics of the functional analysis, and finite element method.