The Finite Element Method, 7.5 Credits

Contents

The course consists of two elements.

Element 1 (4 credits): Theory.
This course focuses on finite element methods (FEM) for the numerical solution of linear and nonlinear partial differential equations (PDE). The most important finite elements are introduced, for example higher order polynomials on tetrahedra and hexahedra as well as isoparametric elements. An abstract framework for the analysis of elliptic problems is used throughout the course, in example to prove existence and uniqueness and for error analysis.

Element 2 (3.5 credits): Computer lab work.
Implementation of FEM and examples of its applications in real world problems is treated in mandatory computer sessions.
 

Expected learning outcomes

For a passing grade, the student must be able to
 
Knowledge and understanding 

• account in detail for, and prove theorems in the abstract framework used for analysis of finite element method

Skills and abilities

• independently formulate, implement and use various finite element methods for linear and non-linear PDEs
• solve the systems of equations resulting from a finite element method in a numerically efficient manner
• derive common error estimates for finite element methods
• use fundamental PDEs in applications
• present results verbally as well as in written form
• show existence and uniqueness for analytical and numerical solutions to elliptic PDEs

Judgment and approach

• evaluate the efficiency of the finite element method numerically

Required Knowledge

The course requires 90 ECTS of which 22,5 ECTS in Calculus including a course in Multivariable Calculus, a course in Linear Algebra on basic level and a course in Numerical Methods for Partial Differential Equations on advanced level or equivalent. Proficiency in English and Swedish equivalent to the level required for basic eligibility for higher studies.

Form of instruction

The teaching in element 1 takes the form of lectures and lessons. The teaching in element 2 takes the form of lab work and seminars.

Examination modes

Element 1 is assessed through a written examination. Element 2 is assessed through seminars and written lab reports. For Element 1, one of the following judgements is awarded: Fail (U), Pass (3), Pass with merit (4), Pass with distinction (5). For Element 2, one of the following judgements is awarded: Fail (U), or Pass (G). For the course as whole, one of the following grades is awarded: Fail (U), Pass (3), Pass with merit (4), Pass with distinction (5). The grade for the whole course is determined by the judgement given for Element 1. To pass the whole course, all elements must have been passed. The grade is only set once all compulsory elements have been assessed.

A student who has been awarded a passing grade for the course cannot be reassessed for a higher grade. Students who do not pass a test or examination on the original date are given another date to retake the examination. A student who has sat two examinations for a course or a part of a course, without passing either examination, has the right to have another examiner appointed, provided there are no specific reasons for not doing so (Chapter 6, Section 22, HEO). The request for a new examiner is made to the Head of the Department of Mathematics and Mathematical Statistics. Examinations based on this course syllabus are guaranteed to be offered for two years after the date of the student's first registration for the course.
 
Credit transfers
Students are entitled to an assessment of whether previous education or equivalent knowledge and skills acquired in professional experience can be accredited for equivalent studies at Umeå University. Applications for credit transfers must be sent to Student Services/Degree Evaluation Office. More information on credit transfers can be found on Umeå University's student website, www.student.umu.se, and in the Higher Education Ordinance (Chapter 6). Rejected applications for credit transfers can be appealed (Higher Education Ordinance, Chapter 12) to the Higher Education Appeals Board. This applies regardless of whether the rejection relates to all or parts of the credit transfer application.
 

Other regulations

In a degree, this course may not be included together with another course with a similar content. If unsure, students should ask the Director of Studies in Mathematics and Mathematical Statistics. The course can also be included in the subject area of computational science and engineering.