A posteriori error estimation in numerical simulation - NNUM054
Title: A posteriorní odhady chyby v numerických simulacích
Guaranteed by: Department of Numerical Mathematics (32-KNM)
Faculty: Faculty of Mathematics and Physics
Actual: from 2018
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: cancelled
Language: Czech
Teaching methods: full-time
Guarantor: prof. Ing. Martin Vohralík, Ph.D.
Classification: Mathematics > Numerical Analysis
Interchangeability : NMNV464
Is incompatible with: NMNV464
Is interchangeable with: NMNV464
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Annotation -
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)
Aim of the course -

The course gives students a knowledge of basics of a posteriori error estimates for various numerical methods.

Last update: T_KNM (05.05.2011)
Literature -

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)
Teaching methods -

Lectures in a lecture hall.

Last update: T_KNM (05.05.2011)
Requirements to the exam -

Examination according to the syllabus.

Last update: T_KNM (05.05.2011)
Syllabus -
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
  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
Last update: T_KNM (12.05.2011)
Entry requirements -

Students are expected to have attended a basic course of functional analysis and to have attended or to attend a course on the modern theory of partial differential equations, e.g., NDIR045.

Last update: T_KNM (05.05.2011)