| Code |
15615
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| Year |
1
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| Semester |
S1
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| ECTS Credits |
8
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| Workload |
TP(60H)
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| Scientific area |
Mathematics
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Entry requirements |
Does not have.
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Learning outcomes |
The main objective of this course is to develop in students the skills necessary to numerically solve problems involving differential equations with boundary values and partial derivatives, and to analyze and interpret the solutions thus obtained. In particular, it is intended that students acquire the theoretical and practical foundations related to finite difference methods, Galerkin and finite element method.
At the end of this course the student should be able to:
- Identify and apply numerical methods appropriate to the problem under study;
- Know the main advantages and disadvantages of the numerical schemes studied;
- Study the consistency and stability of a numerical scheme;
- Computationally implement the different numerical methods.
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Syllabus |
1. Ordinary differential equations with boundary values
1.1. Shooting method
1.2. Collocation method
1.3. Least squares method
1.4. Method of residuals
1.5. Variational formulation
1.6. Finite element method
1.7. Finite difference method
2. Differential equations with partial derivatives
2.1. Stationary problems
2.1.1. Finite difference methods – stability and convergence
2.1.2. Galerkin Methods – variational formulation, Lax-Milgram Theorem, Cea's Lemma
2.1.3. Finite element methods - mesh generation, element spaces, stability and convergence
2.2. Evolutionary Problems
2.2.1. Finite difference methods – stability and convergence
2.2.2. Galerkin methods – variational formulation, Caratheodory's Theorem
2.2.3. Finite element methods – stability and convergence
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Main Bibliography |
Larsson, Stig and Thomée, Vidar Partial differential equations with numerical methods. Texts in Applied Mathematics,45. Springer-Verlag, Berlin, 2003.
Lynch, Daniel R. Numerical Partial Differential Equations for Environmental Scientists and Engineers. Springer US,United States, 2005
Burden, Richard L. and Faires, J. Douglas and Burden, Annette M. Numerical Analysis, Cengage Learning, United States, 2016.
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Teaching Methodologies and Assessment Criteria |
The curricular unit is structured in theoretical-practical classes. The teacher introduces the concepts, states and proves the fundamental results, provides examples and applications. The combination of the theory with the practice in the classes allows the exercises to be performed immediately after each theoretical content, which improves the acquisition of knowledge and skills. In addition, the student is encouraged to participate in classes, to interact with the teacher and with colleagues, and to work autonomously, in the form of exercises, formulation and problem solving.
The evaluation carried out during the teaching-learning process consists of one written test quoted for 12 values and small projects of an analytical and computational nature that involve the application of the studied methods,quoted for 8 values in total. The student can also take a final exam quoted for 20 values.
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Language |
Portuguese. Tutorial support is available in English.
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