Syllabus
* Introductory concepts
Classical solutions, domains with the Lipschitz boundary, Green's theorem, classification of the equations of the second order, Fourier method demonstrated on the scalar wave equation.
* 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.
* Spectral theory
Definition and properties of the Green operator; eigenvalues of the Green operator.
* Finite elements
Basic concepts and ideas of the finite element method. Numerical examples.
* Literature:
M. Křížek, P. Neittaanmaki: Finite Element Approach of Variational Problems and Applications, Longman and J. Wiley & Sons, New York, 1990.
Annotation
Classification of the equations of the second order. Weak formulation of the Dirichlet and the Neumann problem for the elliptic equations, mixed problems.
Basic principles of the numerical solution - finite element method. Evolution equations.