Course syllabus for Partial differential equations

The course syllabus contains changes
See changes

Course syllabus adopted 2019-02-12 by Head of Programme (or corresponding).

Overview

  • Swedish namePartiella differentialekvationer
  • CodeMVE455
  • Credits4.5 Credits
  • OwnerTKKEF
  • Education cycleFirst-cycle
  • Main field of studyMathematics
  • DepartmentMATHEMATICAL SCIENCES
  • GradingTH - Pass with distinction (5), Pass with credit (4), Pass (3), Fail

Course round 1

  • Teaching language English
  • Application code 54118
  • Maximum participants40
  • Open for exchange studentsNo
  • Only students with the course round in the programme overview.

Credit distribution

0115 Project 1.5 c
Grading: UG		 1.5 c
0215 Examination 3 c
Grading: TH		 3 c
  • 15 Mar 2021 am J
  • 09 Jun 2021 pm J
  • 26 Aug 2021 am J

In programmes

Examiner

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Eligibility

General entry requirements for bachelor's level (first cycle)
Applicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling the requirements above.

Specific entry requirements

The same as for the programme that owns the course.
Applicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling the requirements above.

Course specific prerequisites

The participant is presumed to have
(i) a solid background in calculus of one and several variables,
(ii) knowledge of linear algebra/geometry such as vector and matrix algebra and linear spaces,
(iii) knowledge of the elementary theory of linear ordinary differential equations,
(iv) an acquantaince with the complex number system and the complex exponential function,
(v) a solid background in Fourier analysis (especially method of separation of variables).
(v) knowledge of the Galerkin finite element method in 1d and polynomial interpolation in 1d (this corresponds to the material in the finite element part of TMA226).

Aim

The objective is two-fold:
(i) to cover an up-to-date basic theory and
(ii) to introduce some modern approximation tools.

Learning outcomes (after completion of the course the student should be able to)

Passing this course students are supposed to be able to:

i) analyse well-posedness (existence, uniqueness ans stability) issues for linear partial differential equations,
ii) construct finite element discretisations of such equations and perform error analyses, also
iii) implement the finite element method for numerically computing the approximate solutions for partial differential equations of science and engineering.

Content

In the theoretical part, we discuss well-posedness (existence/uniqueness) based on weak formulation and minimization problem (Lax-Milgram and Riesz) leading to the existence of a unique solution for the considered problem and study stability concept for the basic PDEs (Poisson, heat, wave and convection-diffusion equations) in the form of Dirichlet, Neumann and Robin (initial) boundary value problems. To solve time dependent PDEs, we also need to consider the study of initial value problems, such as population dynamics and dynamical systems, in the realm of the ordinary differential equations (ODEs).

In the approximation part we focus on constructing and analysing Galerkin finite element methods (approximation by piecewise polynomials, generalizing the known 1d case to several dimensions) from two points of view.

On one hand we analyse the approximation proceedure, and based on both continuous and discrete weak formulations we can gaurantee the existence of a unique discrete solution and its stability. The convergence analysis is based on interpolation estimates and studied both as a priori (theoretical) and a posteriori (computational) error estimates.

On the other hand we deal with the implementation aspects of  a priori and a posteriori error bounds. We derive, e.g. stiffness-, mass- and convection matrices, and load vectors and eventually reach a linear system of equations to solve numerically. The students are encouraged to use a posteriori error analysis to obtain optimal mesh configurations for concrete problems.

Organisation

The course consists of 27 lecture hourse, 17 exercise hour. KF students are supposed to come to the first lecture, and follow the course TMA372 from week 3. There will be assignments in form of a project that contains implementing the finite element method for a practical problem.

Literature

M. Asadzadeh, An Introduction to the Finite Element Method (FEM) for Differential Equations. (Lecture Notes).

Examination including compulsory elements

Written exam and assignment project.

The course syllabus contains changes

  • Changes to course rounds:
    • 2020-11-23: Examinator Examinator changed from Mohammad Asadzadeh (mohammad) to David Cohen (cohend) by Viceprefekt/adm
      [Course round 1]