The course shows how to estimate a posteriori the error in numerical solution of partial differential equations. A unified framework covering classical numerical methods (finite element method, finite volume method, mixed finite
element method, discontinuous Galerkin method) is presented. The emphasis is on fully computable (guaranteed) estimates and their use for efficient calculation (early stopping of linear and nonlinear solvers, adaptive mesh
refinement, adaptive choice of the time step).
Last update: T_KNM (09.05.2011)
Přednáška se zabývá a posteriorními odhady chyby v numerickém řešení parciálních diferenciálních rovnic. Je představen jednotný rámec
zahrnující klasické numerické metody (metoda konečných objemů, metoda konečných prvků, smíšená metoda konečných prvků, nespojitá
Galerkinova metoda). Důraz je kladen na plně spočítatelné (zaručené) odhady a jejich využití pro efektivní výpočty (včasné zastavení lineárních
a nelineárních řešičů, adaptivní zjemňování sítě, adaptivní volba časového kroku).
Aim of the course -
Last update: T_KNM (05.05.2011)
The course gives students a knowledge of basics of a posteriori error estimates for various numerical methods.
Last update: T_KNM (05.05.2011)
Studenti se seznámí se základy a posteriorních odhadů chyb pro různé numerické metody.
Literature -
Last update: T_KNM (09.05.2011)
Vohralík, M., A posteriori error estimates for efficiency and error control in numerical simulations, lecture notes.
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 review of a posteriori error estimation and adaptive mesh-refinement techniques. Teubner-Wiley, Stuttgart, 1996.
Last update: T_KNM (09.05.2011)
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 review of a posteriori error estimation and adaptive mesh-refinement techniques. Teubner-Wiley, Stuttgart, 1996.
Teaching methods -
Last update: T_KNM (05.05.2011)
Lectures in a lecture hall.
Last update: T_KNM (05.05.2011)
Přednášky v posluchárně.
Requirements to the exam -
Last update: T_KNM (05.05.2011)
Examination according to the syllabus.
Last update: T_KNM (05.05.2011)
Zkouška dle sylabu.
Syllabus -
Last update: T_KNM (12.05.2011)
A posteriori error estimation in numerical simulation 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:
How large is the overall error between the exact and approximate solutions and where is it localized?
How to make the numerical simulation efficient - obtain as good as possible result for as small as possible price (calculation time, memory usage)?
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 classical numerical methods are given.
Course topics
Basic notions of an a posteriori estimate:
guaranteed upper bound
local efficiency
asymptotic exactness
robustness with respect to parameters
evaluation cost
Fundamental physical and mathematical principles and theorems