Computational Partial Differential Equations

The course covers the two main discretization methods (the finite difference method and the finite element method) used in large-scale simulation efforts based on models involving partial differential equations (PDEs). All three basic PDE types (elliptic, parabolic, and hyperbolic) are covered, examples of boundary and initial value problems are given. The properties of the numerical schemes are derived in an exemplary way for linear equations. We also discuss to a certain extent how to solve the large linear and nonlinear algebraic systems arising from the discretization effort. Some other methods (finite volume methods, spectral methods, boundary element methods, etc.) are only briefly mentioned. Part of the course are implementation exercises (C/C++ (preferable) or Matlab) and the use of public domain software.

Participants should have the mathematical background equivalent to three semesters of Engineering and Science Mathematics - attendance of the first year B track (Multivariable Calculus, ODE; Linear Algebra, Fourier Methods, Probability) is highly recommended - or Analysis I/II, Linear Algebra I. In addition, a course in Numerical methods (ESM4A or Numerical Methods I) is prerequisite. Familiarity with C/C++ and Matlab (or Octave) is also assumed.

Topics covered are: Types and examples of PDE and initial/boundary value problems, basic properties (maximum principle, variational principles), consistency and stability of finite difference methods, Galerkin finite element schemes, error versus work estimates, direct and iterative methods for discrete systems, examples of nonlinear problems, the idea of adaptivity.

Lectures:	We 8:15-9:30 in East Hall 8, Fr 11:15-12:30 in East Hall 8
Practical Help:	CLAMV Teaching Labs 112/113 Research I (Tue 7pm/by mutual agreement)

• S. Larsson, V. Thomee: Partial Differential Equations with Numerical Methods, Springer, 2003.

• K. Eriksson, D. Estep, P. Hansbo, C. Johnson: Computational Differential Equations, Cambridge Univ. Press, 1996.
• H. P. Langtangen: Computational Partial Differential Equations, Numerical Methods and Diffpack Programming, 2nd edition, Springer, 2003.
• D. Braess: Finite Elements, 2nd edition, Cambridge Univ. Press, 2001.
• G. Evans, J. Blackledge, P. Yardley: Numerical Methods for Partial Differential Equations, 2nd edition, Springer, 2001
• C. Johnson: Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge Univ. Press, 1987 (out of print, sorry).

The solutions to homework and programming assignments handed out in week X need to be turned in before the Friday lecture in week X+1. Code should be sent in a single email to TBA . You should be able to present and explain homework/code on request. One larger project (due in week 12) is an investigation into a public domain software package assigned to a group of 2-3 students in week 3.

Homework and programming assignments handed in are individual work. This does not prevent you from discussing difficult problems with your peers (or the teacher). If significant help is received from others then this needs to be stated in the submitted solution (as should other sources be quoted). Copying will be considered cheating, and punsihed if severe. Similar rules apply to group assignments.

• The final grade will be computed from the weighted average of percentages of maximal scores with the weights

  Homework:	30%
  Projects:	30%
  Final Exam:	40%

  
  according to the following table:

  Cutoff score:	95%	90%	85%	80%	75%	70%	65%	60%	55%	50%	45%	40%
  IUB Points:	1.0	1.33	1.67	2.0	2.33	2.67	3.0	3.33	3.67	4.0	4.33	4.67