Numerical Methods for PDE's

Organization

Time and venue: Tuesdays 14:00 - 16:45 hrs, Utrecht University, Minnaertbuilding room 021. Usually, lectures during 2 * 45 min + 45 min exercise class.

The final grade will be based on assignments, a computer exercise (slightly modified) and, if this seems necessary, a written exam.

Literature:

Brenner, Susanne C.; Scott, L. Ridgway. The mathematical theory of finite element methods. (English summary) Second edition. Texts in Applied Mathematics, 15. Springer-Verlag, New York, 2002. xvi+361 pp. ISBN: 0-387-95451-1

Braess, Dietrich. Finite elements. (English summary) Theory, fast solvers, and applications in solid mechanics. Translated from the 1992 German edition by Larry L. Schumaker. Second edition. Cambridge University Press, Cambridge, 2001. xviii+352 pp. ISBN: 0-521-01195-7

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;

Lectures

February 5: Johnson Ch. 1
February 12: Brenner Scott, Sections 2.1 - 2.4
February 19: Brenner Scott, Sections 2.5 - 2.8
February 26: Brenner Scott, Sections 1.1 - 1.4
March 5: Brenner Scott, Sections 1.6 - 1.7
March 12: Variational formulation of Poisson's equation with various boundary conditions. Bramble-Hilbert lemma. Material covered by BS 5.1-4, Braess p77-78
March 19: Variational formulation of Biharmonic equation and Brenner Scott, Sections 3.1, 3.2 (until page 75).
March 26: n-simplices and barycentric coordinates (Ciarlet p45-46, or, in 2 dimensions, Brenner Scott p83) and Brenner Scott p75-79.
April 1: Brenner Scott p80-83.
April 8: Brenner Scott p84-85. Transformation lemma (Ciarlet p117-120)
April 15: additional notes
April 21: No lecture
April 29: "additional notes" until Sect. 4.5
May 6: Brenner Scott Sect. 3.5. Additional notes" Sect. 7 (download version May 7)
May 13: "additional notes" Sections 4, 5, 6, 8.
May 20: No lecture
May 27:

Exercises

February 5: Johnson Ch 1: 1.1, 1.3, 1.5 (b)-(e). On page 20, show that $b_i=f(x_i) \int_0^1 \phi_i(x) d x$ when $f$ is linear on the support of $\phi_i$
February 12: Brenner Scott, 2.x: 1, 3, 4, 11, 13, 15
February 19: Brenner Scott, 2.x: 7, 8, 9
February 26: Brenner Scott, 1.x: 1, 3, extra: 1, 2, 3.
March 5: Brenner Scott, 1.x: 20, 8, 30. Additional exercise 4.
March 12: Additional exercises 5, 6, 7, 8.
March 19: Additional exercises 9, 10, 11. Brenner Scott 3.x.1, 10, 17.
March 26: Additional exercise 12. Brenner Scott 3.x.18, 19 (in any case for k=2,3,4)
April 1: Brenner Scott 3.x.14, 27.
April 8: Brenner Scott 3.x.28 (+proof), 6, 13, 15.
April 15: Additional exercises 13, 14, 15.
April 21: No exercise class.
April 29: Exercises 1--7 from "additional notes"
May 6: Prove statements from Brenner Scott Example 3.5.7, Exercises 8--13 from "additional notes"
May 13: Exercises -1, 0, 15, 16 from "additional notes", "additional exercises" 16, 17(a),(c),(d).
May 20: No exercise class.
May 27: