Partial Differential Equations II

Garant předmětu:

Lecturer:

Tutor:

Supervisor:

Department of Technical Mathematics

Synopsis:

The course contains the notion of the weak formulation of the boundary problem for the linear elliptic equation of the 2nd order, the concept of the weak solution.

The essential topics are Lax-Milgram's lemma, the theorem on the existence and uniqueness of a weak solution. Equivalence of the mentioned boundary problem and a problem of finding the minimum of a suitable quadratic functional.

Galerkin and Ritz's method of finding an approximate solution.

Fundamentals of vector field theory, div, rot operators and their properties, Gauss theorem, Stokes theorem, Weyl's vector field decomposition to the sum of the ∇φ field and the rot w field, Helmholz's decomposition, and related issues.

Euler, Stokes, Oseen and Navier-Stokes equations for an incompressible fluid. Classical and weak formulations of the boundary problem in the stationary case and mixed initial-boundary problems in the non-stationary case for the Navier-Stokes equations.

Requirements:

Syllabus of lectures:

The course contains the notion of the weak formulation of the boundary problem for the linear elliptic equation of the 2nd order, the concept of the weak solution.

The essential topics are Lax-Milgram's lemma, the theorem on the existence and uniqueness of a weak solution. Equivalence of the mentioned boundary problem and a problem of finding the minimum of a suitable quadratic functional.

Galerkin and Ritz's method of finding an approximate solution.

Fundamentals of vector field theory, div, rot operators and their properties, Gauss theorem, Stokes theorem, Weyl's vector field decomposition to the sum of the ∇φ field and the rot w field, Helmholz's decomposition, and related issues.

Euler, Stokes, Oseen and Navier-Stokes equations for an incompressible fluid. Classical and weak formulations of the boundary problem in the stationary case and mixed initial-boundary problems in the non-stationary case for the Navier-Stokes equations.

Syllabus of tutorials:

Study Objective:

Study materials:

• K. Rektorys: Variational Methods in Mathematics, Science and Engineering. Springer, Netherlands,1977.

• L. C. Evans: Partial differential equations. Graduate Studies in Mathematics, Vol 19, American Mathematical Society, Second Edition 2010.

• M. Chipot: Elliptic equations. An Introductory cource. Birkhauser Verlag, 2009.

• M. E. Taylor: Partial differential equations I. Applied Mathematical Sciences 115, Springer, 2011.

• R. Temam: Navier-Stokes equations. Theory and numerical analysis. North Holland, 1979, and later issues.

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans: