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|Finite Element Methods
Finite Element Methods
Formulas, inequalities or equations you definitely should
know include: Cauchy-Schwartz and Poincaré inequalities (the latter
with the simple 1D proof, see Ex 2.7 in MGL), Euler forward/backward,
Trapezoidal/Crank-Nicolson rule, Green's formula, quadratures:
Will be provided at the exam (if necessary): trace
inequalities and interpolation estimates. Mathematics handbook
for science and engineering may be used.
Given a general linear ODE on the form My' = -Ay, the
generalized eigenvalue decomposition is a way to diagonalize
it. The result is a set of decoupled scalar ODEs (with the
scalars being in fact minus the eigenvalues of A). [A very
quick derivation now runs as follows: given My' = -Ay, where M = L*L'
(from Cholesky factorization), L'y' = -L'AL*L'y, or with z := L'y and
B := L'AL, z' = -Bz = -UDU'z (from eigendecomposition of B), or with w
:= U'z, w' = -Dw. D is a diagonal matrix with the eigenvalues of B (or
A, they are the same) on the diagonal.]
Some comments on the gain in doing the assignments added under
Assignments and Examination.
Two more exams (from 2010) added under Examination.
Suggested extra material: FEM lectures by Gilbert Strang, FEM
1D part 1, FEM
1D part 2.
There are suggested solutions to most exercises. You find them
under Schedule (follow the links under 'Topics').
IMPORTANT: If you register for the course but decides to
discontinue taking it, be sure to report this fact to the Student
Office email@example.com. The
registration can be removed if you do this within 3 weeks from start.
2010-10-26 The first lecture is in P2446 at 1515.
Is available here.
Language of Instruction
The course consists of 12 lectures, 6 exercise classes, and 3
laborations. Two lectures are given by external visitors and are
important to the learning objectives of the course. These two
lectures are therefore mandatory. The laborations count as 2.0 hp,
while the written exam makes up for the remaining 3.0 hp. There are
three voluntarily assignments: if you submit them before
deadline I will correct them and give feedback.
We will follow reference 1 closely. It is available for free.
Larson, M.G., Bengzon, F.:
The Finite Element Method: Theory, Implementation, and Practice. Department of Mathematics, Umeå University 2009.
Numerical Solution of Partial Differential Equations by
the Finite Element Method. Studentlitteratur, 1987.
Eriksson, K., Estep, D., Hansbo, P. & Johnson, C.
Computational Differential Equations. Studentlitteratur, 1997.