Numerical Methods for Partial Differential Equations
Numeriska metoder för partiella differentialekvationer

FMNN45, 7.5 credits, A (Second Cycle)

Valid for: 2025/26
Faculty: Faculty of Engineering LTH
Decided by: PLED F/Pi
Date of Decision: 2025-04-10
Effective: 2025-05-05

General Information

Depth of study relative to the degree requirements: Second cycle, in-depth level of the course cannot be classified
Elective for: F4, F4-bs, Pi4-bs
Language of instruction: The course will be given in English

Aim

The course is a continuation of FMNN10 Numerical Methods for Differential Equations and the purpose of the course is to deepen the student's knowledge of partial differential equations, with an emphasis on numerical approximation of solutions, and to provide practical training in solving relevant computational problems in a python environment with tools in DUNE (Distributed and Unified Numerics Environment), a modular numerical toolbox for partial differential equations.

Learning outcomes

Knowledge and understanding
For a passing grade the student must

• be able to understand how concepts from functional analysis are used to develop and analyse numerical algorithms for elliptic partial differential equations,
• be able to explain how partial differential equations may be discretized using the Finite Element Method,
• be able to state and derive simple error estimates for elliptic problems,
• be able to independently identify an appropriate numerical method and select suitable parameters with regard to accuracy and efficiency requirements,
• be able to give examples of important fields of applications in which the algorithms occurring in the course are important,
• be able to explain concepts presented in the course with adequate terminology, in writing and orally.

Competences and skills
For a passing grade the student must

• be able to rewrite problem on finite element form and apply suitable solution strategies using DUNE-FEM,
• be able to independently proceed from observation and interpretation of results to conclusion, and present the conclusions on a scientific basis in a free report format,
• be able to independently present results and conclusions of numerical experiments performed with DUNE-FEM, in written and oral form, with references and other documentation of work carried out in support of their conclusions. 

Judgement and approach
For a passing grade the student must

• be able to independently evaluate obtained numerical results with DUNE-FEM in relation to the available theory,

Contents

Theory for ellipic partial differential equations (PDEs): Well-posed problems. Weak theory of solvability. Existence and uniqueness. Stability with respect to data, and approximation of solutions. Regularity of solutions.

Solution with the Finite Element Method in DUNE-FEM: Introduction to DUNE-FEM. Discretization of boundary value problems for PDEs in DUNE-FEM using the Finite Element Method. Construction of finite elements, e.g. discretization grids, reference elements, degree of freedom (DOF) mappings.

Using the Unified Form Language (UFL) in DUNE-FEM for description of weak forms of PDEs,

Parallelization of Finite Element methods using domain decomposition,

Adaptive Finite Elements.

Examination details

Grading scale: TH - (U, 3, 4, 5) - (Fail, Three, Four, Five)
Assessment:

A number of compulsory projects carried out in small groups are included in the course. The examination consists of a written final report and an appurtenant oral presentation of the group's solutions of the projects at the end of the course.

Attendance at all oral group presentations of the project results is mandatory.

The examiner, in consultation with Disability Support Services, may deviate from the regular form of examination in order to provide a permanently disabled student with a form of examination equivalent to that of a student without a disability.

Modules
Code: 0125. Name: Numerical Methods for Partial Differential Equations.
Credits: 7.5. Grading scale: TH - (U, 3, 4, 5).

Admission

Admission requirements:

• (FMAB30 Calculus in Several Variables or FMAB35 Calculus in Several Variables) and FMNN10 Numerical Methods for Differential Equations

Reading list

• Larsson, Stig. author: Partial Differential Equations with Numerical Methods / by Stig Larsson, Vidar Thomee [Elektronisk resurs]. Springer, 2009, ISBN: 9783540887065.
• Renardy, Michael: An introduction to partial differential equations [Elektronisk resurs]. New York : Springer, 2004, ISBN: 0387004440.
• Brenner, Susanne C: The Mathematical Theory of Finite Element Methods [elektronisk resurs]. New York, NY : Springer, 2008, ISBN: 9780387759340.
• Ciarlet, Philippe G: The finite element method for elliptic problems. Philadelphia : Society for Industrial and Applied Mathematics (SIAM, 2002, ISBN: 0898715148.
• Thomée, Vidar: Galerkin Finite Element Methods for Parabolic Problems [electronic resource]. Berlin Heidelberg : Springer-Verlag GmbH, 2006, ISBN: 9783540331223.

Contact

Teacher: Eskil Hansen, Eskil.Hansen@math.lth.se
Teacher: Robert Klöfkorn, Robert.Klofkorn@math.lu.se
Examinator: Robert Klöfkorn, Robert.Klofkorn@math.lu.se
Director of studies: Anders Holst, Anders.Holst@math.lth.se