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Lecturer(s)
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Machalová Jitka, doc. RNDr. Ph.D.
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Course content
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1.Brief history of variational calculus and variational methods, their importance for applications. Lagrange's principle of minimum potential energy. 2.Fundamental concepts and propositions of variational calculus. Differentiability in the sense of Gateaux. Convexity, weak lower semi-continuity. Quadratic functional and its properties. 3.Variational calculus problems formulations. Fundamental theorems of variational calculus. Coercivity of a functional. Results of the classical variational calculus. Euler-Lagrange equation. 4.Variational formulations of elliptic boundary value problems of the 2nd order. Functionals for Dirichlet, Neumann, Newton and mixed boundary problems. Relationship between classical and variational formulation. Linear elasticity problem. 5.Solvability of elliptic boundary value problems of the 2nd order. Solution analysis of the variational formulation for Dirichlet, Neumann, Newton and mixed problem in terms of existence and uniqueness. 6.Selected elliptic boundary value problems of the 4th order and their variational formulations. Problem in 1D: bending of Euler-Bernoulliho beam. Problem in 2D: biharmonic equation, bending of a thin plate. 7.Nonsymmetric elliptic problems and their solutions. Problems formulations and solvability for a nonsymmetric elliptic operator of the 2nd order. 8.Elliptic variational inequalities. Solvability conditions. Elliptic variational inequalities of the first and second kind. Outer normal derivative. Examples. 9.Nonsymmetric variational inequalities. Abstract variational inequality. Lions-Stampacchia theorems about solution existence. Some applications of variational inequalities. 10.Ritz method for variational problems. Principle of the method, convergence conditions and error estimation. Choice of basis functions. Examples. 11.Galerkin method for variational equations. Principle of the method, convergence conditions and error estimation. Examples. A short overview of history. 12.Ritz-Galerkin method for solution of variational inequalities. Principles of its applications and differences in comparison to variational equalities. Convergence analysis for. variational inequalities of the first and second kind.
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Learning activities and teaching methods
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Lecture, Monologic Lecture(Interpretation, Training), Dialogic Lecture (Discussion, Dialog, Brainstorming)
- Attendace
- 39 hours per semester
- Homework for Teaching
- 15 hours per semester
- Preparation for the Course Credit
- 15 hours per semester
- Preparation for the Exam
- 50 hours per semester
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Learning outcomes
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Lesson 1 Brief history of variational calculus and variational methods, their importance for applications. Lagrange's principle of minimum potential energy. Lesson 2 Fundamental concepts and propositions of variational calculus. Differentiability in the sense of Gateaux. Convexity, weak lower semi-continuity. Quadratic functional and its properties. Lesson 3 Variational calculus problems formulations. Fundamental theorems of variational calculus. Coercivity of a functional. Results of the classical variational calculus. Euler-Lagrange equation. Lesson 4 Variational formulations of elliptic boundary value problems of the 2nd order. Functionals for Dirichlet, Neumann, Newton and mixed boundary problems. Relationship between classical and variational formulation. Linear elasticity problem. Lesson 5 Solvability of elliptic boundary value problems of the 2nd order. Solution analysis of the variational formulation for Dirichlet, Neumann, Newton and mixed problem in terms of existence and uniqueness. Lesson 6 Selected elliptic boundary value problems of the 4th order and their variational formulations. Problem in 1D: bending of Euler-Bernoulliho beam. Problem in 2D: biharmonic equation, bending of a thin plate. Lesson 7 Nonsymmetric elliptic problems and their solutions. Problems formulations and solvability for a nonsymmetric elliptic operator of the 2nd order. Lesson 8 Elliptic variational inequalities. Solvability conditions. Elliptic variational inequalities of the first and second kind. Outer normal derivative. Examples. Lesson 9 Nonsymmetric variational inequalities. Abstract variational inequality. Lions-Stampacchia theorems about solution existence. Some applications of variational inequalities. Lesson 10 Ritz method for variational problems. Principle of the method, convergence conditions and error estimation. Choice of basis functions. Examples. Lesson 11 Galerkin method for variational equations. Principle of the method, convergence conditions and error estimation. Examples. A short overview of history. Lesson 12 Ritz-Galerkin method for solution of variational inequalities. Principles of its applications and differences in comparison to variational equalities. Convergence analysis for. variational inequalities of the first and second kind.
Application Apply calculus of variations and optimization theory to boundary value problems in order to obtain some solution methods.
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Prerequisites
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Standard knowledge from mathematical and functional analysis. Information concerning optimization and partial differential equations are welcomed, but not necessary.
KMA/PDR2
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Assessment methods and criteria
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Oral exam
Credit: active participation in seminars. Exam: the student has to understand the subject and be acquainted with theoretical and practical aspects of the method.
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Recommended literature
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J. Cea. (1971). Optimization. Theorie et algorithmes. Dunod, Paris.
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J. Haslinger. (1980). Metoda konečných prvků pro řešení eliptických rovnic a nerovnic. SPN, Praha.
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J. Machalová, H. Netuka. (2014). Variační metody. Univerzita Palackého v Olomouci, Olomouc.
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K. Rektorys. (1974). Variační metody v inženýrských problémech a v problémech matematické fyziky. SNTL, Praha.
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K. Rektorys. (2002). Variational methods in mathematics, science and engineering. Springer.
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S.C. Brenner, L.R. Scott. (2008). The mathematical theory of finite element methods. Springer.
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