Syllabus for Advanced Numerical Methods

Avancerade numeriska metoder

Syllabus

  • 10 credits
  • Course code: 1TD050
  • Education cycle: Second cycle
  • Main field(s) of study and in-depth level: Computational Science A1F, Computer Science A1F, Technology A1F
  • Grading system: Fail (U), Pass (3), Pass with credit (4), Pass with distinction (5)
  • Established: 2016-03-10
  • Established by:
  • Revised: 2022-10-20
  • Revised by: The Faculty Board of Science and Technology
  • Applies from: Autumn 2023
  • Entry requirements:

    120 credits in science/engineering including 45 credits in mathematics, where linear algebra, vector calculus, transform theory (Fourier analysis) must be covered. Scientific Computing III or Scientific computing for Partial Differential Equations. Applied Finite Element Methods or Finite Element Methods. Proficiency in English equivalent to the Swedish upper secondary course English 6.

  • Responsible department: Department of Information Technology

Learning outcomes

On completion of the course, the student should be able to:

  • review and apply fundamental theory for mathematical modelling with partial differential equations;
  • analyse finite difference and finite element approximations of systems of partial differential equations;
  • review and describe application areas where different types of finite element and finite differences are used;
  • choose, formulate and implement appropriate numerical methods for solving science and engineering problems that are formulated as partial differential equations;
  • interpret, analyse and evaluate results from numerical computations;
  • use common software to solve application problems formulated as more complicated partial differential equations, such as linear elasticity and transport problems.

Content

The content is built up around a design problem. It focuses on keywords such as consistency, convergence, stability, existence, uniqueness and efficiency.
The course covers the Fourier method, Energy method, normed vector spaces, bilinear forms, Lp - and Sobolev spaces, weak derivatives, elliptic boundary value problems, hyperbolic and parabolic time-dependent initial value problems, linearisation for nonlinear problems, interpolants and finite elements, higher order methods and elements, stabilisation, error bounds.
Analyse linear systems of partial differential equations, analyse finite difference and finite element methods of systems of nonlinear partial differential equations. The methodology in finite difference and finite element methods (Lax-Richtmeyer, Lax-Milgram) and applying explicit and implicit time discretisation methods.

Instruction

Lectures, problem solving sessions and assignments.

Assessment

Written final exam or, if less than 10 participants, oral exam (6 credits). Assignments (4 credits).

If there are special reasons for doing so, an examiner may make an exception from the method of assessment indicated and allow a student to be assessed by another method. An example of special reasons might be a certificate regarding special pedagogical support from the disability coordinator of the university.

Reading list

Reading list

Applies from: Autumn 2023

Some titles may be available electronically through the University library.

  • Larson, M. G.; Bengzon, F. A first course in finite elements: lecture notes

    Department of Mathematics, Umeå university, 2010

    Mandatory

Last modified: 2022-04-26