At the end of the course,
- You will be able to construct and use finite elements to solve partial differential equations.
- You will be able to analyze finite element solutions in terms of their stability, convergence, and approximation properties.
- You will be able to convert a finite element algorithm into a well-designed code.
- You will know how to test (verify) a numerical simulation code.
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Partial differential equations (PDEs) describe the laws of nature around us. They appear as mathematical models in a wide variety of physical contexts, from the evolution of heat over time to the fluid dynamics, and the propagation of sound waves. In reality, they are too complicated to be solved exactly, so numerical methods are essential. This course will introduce you to one of the most popular numerical techniques for solving PDEs: the finite element method. We will combine learning about the mathematical aspects of finite elements, such as stability and accuracy, with their practical implementation.
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The course is largely self-contained. The needed theoretical basics (mainly of Sobolev spaces) will be introduced at the beginning. If you are unfamiliar with them, additional self-study is recommended for a deeper understanding of the material.
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Exam, which carries 80% of the final grade and will be written or oral depending on the number of participants, and a coding project, which carries 20% of the final grade. The coding project will have a fixed deadline and no resit opportunity.
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