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Syllabus for

Academic year
TMA372 - Partial differential equations, first course
Partiella differentialekvationer, grundkurs
 
Syllabus adopted 2019-02-22 by Head of Programme (or corresponding)
Owner: MPENM
7,5 Credits
Grading: TH - Pass with distinction (5), Pass with credit (4), Pass (3), Fail
Education cycle: Second-cycle
Major subject: Mathematics
Department: 11 - MATHEMATICAL SCIENCES


Teaching language: English
Application code: 20141
Open for exchange students: Yes

Module   Credit distribution   Examination dates
Sp1 Sp2 Sp3 Sp4 Summer course No Sp
0101 Examination 7,5c Grading: TH   7,5c    

In programs

MPENM ENGINEERING MATHEMATICS AND COMPUTATIONAL SCIENCE, MSC PROGR, Year 1 (compulsory elective)
TKELT ELECTRICAL ENGINEERING, Year 3 (compulsory elective)
TKTEM ENGINEERING MATHEMATICS, Year 3 (elective)
TKAUT AUTOMATION AND MECHATRONICS ENGINEERING, Year 3 (elective)

Examiner:

Mohammad Asadzadeh

  Go to Course Homepage


Eligibility

General entry requirements for Master's level (second 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

English 6 (or by other approved means with the equivalent proficiency level)
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

A solid background in modern linear algebra and calculus in one and several variables. A solid background in Fourier analysis, especially the method of separation of variables for solving PDEs.

Aim

This course gives an introduction to the modern theory of partial differential equations (PDEs) with applications in science and engineering. It also presents an introduction to the finite element method as a general tool for numerical solution of PDEs.

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

  • derive weak formulations of the basic initial-boundary value problems for PDEs.
  • derive stability estimates for the continuous problems and predict the influence of data.
  • formulate Galerkin finite element methods for PDEs and dynamical systems.
  • derive error estimates using exact solution (a priori) and numerical solution (a posteriori).
  • explain how the finite element method is implemented in computer code.
  • improve the error estimates modifying the method or employing adaptive procedure.
  • draw relevant conclusions about stability, reliability and efficiency of the methods.

Content

Weak solutions to elliptic, parabolic, and hyperbolic partial differential equations (PDE). Computation of approximate solutions to various PDE by the finite element method (as well as dynamical systems). Interpolation, quadrature and linear systems. A brief introduction to representation theorems and
abstract theory to justify the weak (variational) approach. A priori and a posteriori error estimates. Applications to e.g. diffusion, heat conduction, and wave propagation.

More precisely the course covers following topics:

Basic interpolation theory:
Interpolation with polynomials
Interpolations error analys
quadrature rules and quadrature error.

Numerical linear algebra:
Solving linear system of equation with
Jacobi's method
Gauss- Seidel and
Overrelaxation methods.
Dynamical system:
Structures in approximation with polynomials
Ill-conditioned system.

Finite element method for boundary-value problem in 1D:
Stability
Error estimates and algorithms
Finite element method för initial-value problem i 1D:
Fundamental solution
Stability
Error estimates and algorithms
The dual problem.
Lax-Milgram theorem:
Abstract formulation
Riesz representation theorem
Studies and Analysis of problems in higher space dimensions:
Finite element in higher space dimensions.
Finite element method for Poisson equation in higher space dimensions.
Finite element method for heat equation in higher space dimensions.
Stability
Error estimates and numerical algorithms
Finite element method for wave equation in higher space dimensions.
Fundamental solution
Stability
Error estimates and numerical algorithms
Finite element method for convection-diffusion equations:
Stability
Error estimates and numerical algorithms

Organisation

Lectures (about 35 hours), exercises (about 21 hours) and home assignments consisting of both theoretical and computer assingments.

Literature

M. Asadzadeh : An Introduction to Finite Element Methods (FEM) for Differential Equations. (Available in Cremona)

M. Asadzadeh, Lecture Notes in PDE (electronic)

Examination including compulsory elements

Home and computer assignments combined with written exam


Published: Mon 28 Nov 2016.