2821O: TOPICS IN A POSTERIORI ERROR ESTIMATIONS: FINITE ELEMENT & REDUCED BASIS METHOD

Office: 180 George, Room 18, Office Hour: By Appointment

Time/Location: Monday, 12:30-2:50 at 102A, 180 George

Course Description: The course will contain two related parts.

1) An introduction of different types of a posteriori error estimations for various finite element methods. 2) An introduction of certified reduced basis method, where a posteriori error estimations play an important role.

The course will emphasize both the theory and implementation. Related Matlab programs for both FEM and RBM will be provided. Residual-type, local-problem type, and recovery-type error estimators for conforming, mixed, non-conforming, and discontinuous galerkin finite element methods for different types of equations will be studied.

For reduced basis methods, offline-online procedure, greedy algorithm, error estimator, empirical interpolation method, and successive constraint method will be discussed.

Goal-Oriented primal-dual approach for both FEM and RBM will be covered. Objective: To learn various theoretical and practical results of adaptive finite element methods and reduced basis methods.

Grading: No exams or test. The grades will be based on two sets of problems, one set for adaptive finite elements and one set for reduced basis methods. Each set of problems will contains 2 or 3 theoretical problems and a matlab programming problem. The grades for two sets of problems are going to be equally weighted. Each set of problems will be due two or three weeks after the date it handed out. The last 2-3 weeks of course will not be included in the final grade.

Main Course Material

Short Matlab Finite Element Implementations for Conforming Elements ( pdf , matlab code ) and Mixed Elements ( pdf , matlab code ) ,
(see C. Carstensen's homepage, software)

iFEM by Long Chen at UCI, ( pdf , matlab code download) ,

Adaptive Finite Element Methods by R. Verfurth, pdf

Primer of Adaptive Finite Element Methods by Ricardo H. Nochetto and Andreas Veeser, pdf

Finite Element Methods by S. Brenner and C. Carstensen, pdf

Reduced Basis Methods at MIT, website , book by A.T. Patera and Rozza

Certified reduced basis approximation for parametrized partial differential equations and applications by Alfio Quarteroni,Gianluigi Rozza1, and Andrea Manzoni,Journal of Mathematics in Industry 2011, 1:3, pdf

RB in matlab zip dowload

Other Reference

A Review of Posteriori Error Estimation & Adaptive Mesh-Refinement Techniques by Rudiger Verfurth, 1996

A posteriori error estimation in finite element analysis by Mark Ainsworth and John Tinsley Oden, Willey, 2000.

A posteriori error estimation in finite element analysis by Mark Ainsworth and John Tinsley Oden,Computer Methods in Applied Mechanics and Engineering Volume 142, Issues 1Ð2, 15 March 1997, Pages 1Ð88, pdf dowload


Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics by D. Braess, Cambridge U Press