One week course - "The Weak Formulation of Partial Differential Equations"

ONE-WEEK COURSE ON THE WEAK FORMULATION OF PARTIAL DIFFERENTIAL EQUATIONS

DATE: November 28 - December 2, 2005

PLACE: Department of Geophysics of the Charles University, Prague

LECTURER: Ctirad Matyska

SYLLABUS:
Introductory concepts:
Classical solutions, domains with the Lipschitz boundary, Green's theorem, classification of the equations of the second order.

Sobolev spaces:
Definition of the Sobolev space W1,2 , trace theorem, Rellich's theorem.

Linear elliptic equations:
Weak and variational formulations.

Dirichlet's problem:
Formulation and interpretation of the weak solution,
Lax-Milgram theorem and uniqueness of the problem,
variational approach - differentiating in the Gateaux sense of the functional of potential energy, sufficient conditions for the existence of the minimum, generalized problem for elliptic equations - existence and uniqueness. Neumann's problem and equilibrium conditions.

Nonlinear equations:
Strictly monotone operators and contraction theorem, uniqueness of the solution.

Finite elements:
Basic concepts and ideas of the finite element method. Numerical examples.

Recommended reading:
K. Rektorys: Variational methods in Mathematics, Science and
Engineering, Kluwer 2001.
M. Krizek, P. Neittaanmaki: Finite Element Approximation of Variational Problems and Applications, Longman and J. Wiley & Sons, New York, 1990.

Contact:
Prof. C. Matyska
cm@karel.troja.mff.cuni.cz