Course syllabus - Finite Element Method

Objectives

The objectives of the course are to give opportunities for students to get an introduction to the finite element method, with a focus on the basic mathematical principles and the computer implementation using the numerical software package MATLAB.

Learning outcomes

Upon completion of the course, students are expected to be able to
1. explain basic principles of elliptic, parabolic, hyperbolic partial differential equations and their classical mathematical models
2. derive error estimates for piecewise polynomial interpolation in one and two dimensions
3. state the variational formulation for a class of linear partial differential equations
4. derive apriori and aposteriori error estimates and construct adaptive finite element methods
5. solve elliptic, parabolic, and hyperbolic partial differential equations by the finite element method
6. implement the numerical methods covered in the course
7. compose structured reports while clearly presenting problem statement, methodology, analysis and assessment of results

Course content

- Definition of elliptic, hyperbolic and parabolic PDE. Definition of well-ordered problems. Dirichlet-, Neumann- and Robin boundary
- Value conditions. Discrete function spaces in one and in two dimensions. Norm and scalar product in discrete function spaces
- Variational formulations of elliptic boundary-value problems
- Finite element method in one and in two dimensions. Maximum principle, the discrete and the continuous. Error estimates for approximations by finite element method (FEM) of elliptic problems. Mesh division and adaptive mesh refinement. Implementation of FEM in MATLAB. Utilization of FEM-software

Specific requirements

At least totally 60 credits in the engineering, natural sciences, business administration or economics areas including Basic Vector Algebra, 7.5 credits, Single Variable Calculus, 7.5 credits, Numerical Methods with Matlab, 7.5 credits, out of which 2.5 credits must be completed at the beginning of the course, and Calculus of Several Variables, 7.5 credits, out of which 3.5 credits must be completed at the beginning of the course, or equivalent.

Examination

TEN1, Written examination, 4 credits, individual written examination concerning learning outcomes 1-5, grades Fail (U), Pass (G) or Pass with distinction (VG).
LAB1, Laboratory work, 3.5 credits, individual and/or group laboratory work concerning learning outcomes 1-7, grades Fail (U) or Pass (G).

A student who has a certificate from MDU regarding a disability has the opportunity to submit a request for supportive measures during written examinations or other forms of examination, in accordance with the Rules and Regulations for Examinations at First-cycle and Second-cycle Level at Mälardalen University (2020/1655). It is the examiner who takes decisions on any supportive measures, based on what kind of certificate is issued, and in that case which measures are to be applied.

Suspicions of attempting to deceive in examinations (cheating) are reported to the Vice-Chancellor, in accordance with the Higher Education Ordinance, and are examined by the University’s Disciplinary Board. If the Disciplinary Board considers the student to be guilty of a disciplinary offence, the Board will take a decision on disciplinary action, which will be a warning or suspension.

Grade

Pass with distinction, Pass, Fail