Numerical Methods for stationary PDEs (spring 2012)
Organization
Time and venue: Wednesdays 10:15 - 13:00 hrs in weeks 6-21, Vrije Universiteit, W&N Building. Usually, lectures during 2 * 45 min + 45 min exercise class.
The final grade will be based on assignments (2/3) (lowest two grades will be ignored) and a
computer exercise (1/3) (deadline June 6).
Literature:
Brenner, Susanne C.; Scott, L. Ridgway.
The mathematical theory of finite element methods.
Second edition. Texts in Applied Mathematics, 15. Springer-Verlag, New York, 2002. xvi+361 pp. ISBN: 0-387-95451-1 (or the third edition from 2008), the book.
Braess, Dietrich.
Finite elements. Theory, fast solvers, and applications in elasticity theory. Translated from the German by Larry L. Schumaker. Third edition. Cambridge University Press, Cambridge, 2007. xviii+365 pp. ISBN: 978-0-521-70518-9
Ciarlet, Philippe G.
The finite element method for elliptic problems.
North-Holland, Amsterdam,1978
Johnson, Claes.
Numerical solution of partial differential equations by the finite element method. Cambridge University Press, Cambridge, 1987. 278 pp. ISBN: 0-521-34514-6;
Ricardo H. Nochetto and Andreas Veeser.
Primer of Adaptive Finite Element Methods
Additional notes
Lectures
- February 8 (room P647): Classification of second order PDEs into elliptic, parabolic and hyperbolic. Model examples: Poisson, heat and wave equation. Book: pp 1-5.
- February 15 (P647): Sect. 0.3, 0.4, 1.1, 1.2
- February 22 (P647): Sect. 1.3-7
- February 29 (P647): Sect. 2.1-6
- March 7 (P647): Sect. 2.7-9, Sect. 3.1 apart from its last lemma
- March 14 (P647): until (3.3.8) in Section 3.3
- March 21 (P647): Ch. 3 until Definition 3.4.6.
- March 28 (C121): remainder Ch. 3, Bramble-Hilbert lemma from ``additional notes''
- April 4 (P647): "additional notes" until Thm. 3.6
- April 11 (P647): Sect. 5.1-5.4
- April 18 (P647): Quasi-interpolators (Sect. 4.8)
- April 25 (P647): more or less the material from Sect. 9.2 and 9.3 (third edition)
- May 2: No lecture
- May 9 (P647): nonlinear approximation, potential of adaptivity, AFEM algorithm, newest vertex bisection
- May 16 (P647): convergence and optimality of AFEM
- May 23 (P647): bounding the cardinality of a partition on a multiple of the number of elements that have been marked for refinement
Additional exercises
Exercises
(in boldface those that have been graded)
- February 8 (room P647): 0.x.2, 0.x.3, 0.x.8
- February 15 (P647): extra 1, 0.x.6, 1.x.1, 1.x.10
- February 22 (P647): 1.x.4, 1.x.5, 1.x.8 (note that you simply can take a classical derivative), 1.x.13 (first do it on the unit ball without a small ball around zero)
- February 29 (P647): 1.x.16, 1.x.20, 1.x.35, 2.x.10 + Show that for any k, the differential equation from 2.x.10, but with boundary conditions u(0)=u(1)=0, leads to a coercive bilinear form over the space V from 2.x.12
- March 7 (P647): 2.x: 7, 8, 9. 3.x: 10, 30.
- March 14 (P647): extra 2, 3.x: 17, 18, 19.
- March 21 (P647): 3.x: 14 (for the second question it is sufficient to show that the points can be chosen so that things go wrong), 27 (sufficient to construct a counterexample for a subdivision consisting of two triangles), 28 , 6.
- March 28 (C121): 3.x: 8, 9, 15. extra 3
- April 4 (P647): extra 4, 11 (use book, Thm, 1.6.6), 5.x.2
- April 11 (P647): 5.x.1, extra 6, 7.
- April 18 (P647): extra 5, 12, 9.x.5, 9.x.6.
- April 25 (P647): extra (version of April 29 or later) 8, 13
- May 2: No lecture
- May 9 (P647): extra 14, 15
- May 16 (P647): extra 16, 17
- May 23 (P647): no new exercises