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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.
Last update: T_KG (16.05.2001)
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The lecture is devoted to introduction into the weak formulation of equations of mathematical physics and their numerical solution. Last update: T_KG (26.03.2008)
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Zkouška je ústní, požadavky odpovídají sylabu v rozsahu prezentovaném na přednášce. Last update: Matyska Ctirad, doc. RNDr., DrSc. (11.10.2017)
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Last update: Zakouřil Pavel, RNDr., Ph.D. (05.08.2002)
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Lecture Last update: T_KG (11.04.2008)
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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. Last update: T_KG (07.05.2002)
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