MM834: Partial differential equations: theory, modelling and simulation

Study Board of Science

Teaching language: Danish or English depending on the teacher, but English if international students are enrolled
EKA: N310005102
Censorship: Second examiner: Internal
Grading: 7-point grading scale
Offered in: Odense
Offered in: Autumn
Level: Master's level course approved as PhD course

STADS ID (UVA): N310005101
ECTS value: 10

Date of Approval: 25-04-2019

Duration: 1 semester

Version: Approved - active


13016001(former UVA) is identical with this course description. 
The course is co-read with MM546.

Entry requirements

A Bachelor’s degree in mathematics, physics or computer science.
The course cannot be chosen by students, who passed MM546.

Academic preconditions

Students taking the course are expected to:

  • Have knowledge of calculus, linear algebra, real analysis, integration theory and Banachspaces.
  • Be able to use python scripting and numerical methods to solve algebraic and ordinary differential equations.

Participant limit


Course introduction

The aim of the course is introduce modeling of problems from science and
engineering by partial differential equations. To analyze and solve
these equations both by analytic tools (when appropriate) and by
computational methods.

The course builds on the knowledge acquired in
the courses MM536 (Calculus for mathematics), MM538 (Algebra and linear
algebra), MM533 (Mathematical and Numerical Analysis), MM547 (Ordinary
differential equations: theory, modelling and simulation), and MM548
(measure- and integration theory, Banachspaces).

The course is of
high multidisciplinary value and gives an academic basis for a Master
Project in several core areas of Natural Sciences.

In relation to the competence profile of the degree it is the explicit focus of the course to:

  • Give the competence to :
    1. handle complex and development-oriented situations in study and work contexts.
    2. Identify needs and plan individual learning
  • Give skills to:
    1. analyze practical and theoretical problems using numerical simulations based on suitable mathematical models.
    2. Analyze qualitative and quantitative properties of mathematical models.
    3. Explain and evaluate errors in modeling and simulation
    4. Explain and select relevant analytic and solution models
    5. Formulate and present problems and solutions to fellow students and partners
  • Give knowledge and understanding of:
    1. mathematical modelling and numerical analysis of problems in natural science and engineering.
    2. theory, methods and praxis within applied mathematics.

Expected learning outcome

The learning objectives of the course are that the student demonstrates the ability to:

  • Be able to deal with partial differential equation models for complex processes in science.
  • Classify 2nd order PDEs and describe their characteristic properties.
  • Analyze and simulate partial differential equations using appropriate, advanced methods and modern software.
  • Construct, implement and analyze numerical methods to compute (approximate) solutions to partial differential equations.
  • Understand the mathematical theory for numerical methods for PDEs.
  • Design and perform reliable simulations of PDE models for complex processes in science.
  • Give a seminar presentation of the individual project and answer supplementary questions.


The following main topics are contained in the course:

  • Classification of 2nd order PDEs: Elliptic, parabolic and hyperbolic problems.
  • Elliptic boundary value problems and Galerkin Finite Elements.
    • Variational formulation, ellipticity, and the Lax-Milgram theorem.
    • Sobolev spaces, Cauchy-Schwarz and Poincare inequalities.
    • The poisson equation: Variational form, ellipticity, and FEniCS implementation.
    • Galerkin's method, Galerkin orthogonality, best approximation, and error analysis.
    • Finite elements for the Poisson equation, error bounds by duality.
    • Neumann, Dirichlet, and Robin boundary conditions.
    • div-grad operators and FEniCS.
  • Parabolic PDEs: The heat equation.
    • Runge-Kutta time stepping in variational form.
    • SDIRK methods and L-stability.
    • Simulation of the heat transfer.
  • Parabolic-elliptic systems: Navier-Stokes equations
    • Chorin’s projection method.
    • Incremental pressure correction – IPC method
    • Simulation of incompressible flow with heat transfer
  • Adaptive calibration of PDE models


See Blackboard for syllabus lists and additional literature references.

Examination regulations

Exam element a)




Project assignment with oral presentation




Second examiner: Internal


7-point grading scale


Student Identification Card


Normally, the same as teaching language

Examination aids

To be announced during the course

ECTS value


Additional information

Reexam in the same exam period or immediately thereafter.The examination form for re-examination may be different from the exam form at the regular exam.

Indicative number of lessons

80 hours per semester

Teaching Method

Activities during the study phase:
  • preparation of exercises in study groups
  • preparation of projects

Teacher responsible

Name E-mail Department
Achim Schroll


08 - 09
09 - 10
10 - 11
11 - 12
12 - 13
13 - 14
14 - 15
15 - 16
Show full time table

Administrative Unit

Institut for Matematik og Datalogi (matematik)

Team at Registration & Legality


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