SF2561 The Finite Element Method 7.5 credits

Finita elementmetoden

A second course on computational methods focusing on the finite element method (FEM) and partial differential equations.

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Course information

Content and learning outcomes

Course contents *

  • FEM-formulation of linear and non-linear partial differential equations, element
    types and their implementation, grid generation, adaption and error control, efficient
    solution algorithms (e.g. by a multigrid method).
  • Applications to stationary and transient diffusion processes, elasticity, convectiondiffusion,
    Navier-Stokes equation, quantum mechanics etc

Intended learning outcomes *

Basic laws of nature are typically expressed in the form of partial differential equations (PDE), such as Navier’s equations of elasticity, Maxwell’s equations of electromagnetics,Navier-Stokes equations of fluid flow, and Schrödinger’s equations of quantum mechanics. The Finite element method (FEM) has emerged as a universal tool for the computational solution of PDEs with a multitude of applications in engineering and science. Adaptivity is an important computational technology where the FEM algorithm is automatically tailored to compute a user specified output of interest to a chosen accuracy, to a minimal computational cost.

This FEM course aims to provide the student both with theoretical and practical skills, including the ability to formulate and implement adaptive FEM algorithms for an important family of PDEs.

The theoretical part of this course deals mainly with scalar linear PDE, after which the student will be able to

  • derive the weak formulation
  • formulate a corresponding FEM approximation;
  • estimate the stability of a given linear PDE and it’s FEM approximation;
  • derive a priori and a posteriori error estimates in the energy norm, the L2-norm,
    and linear functionals of the solution;
  • state and use the Lax-Milgram theorem for a given variational problem.

Having completed the practical part of the course the student will be able to:
modify an existing FEM program to solve a new scalar PDE (possibly nonlinear);

  • implement an adaptive mesh refinement algorithm, based on an a posteriori error
  • estimate derived in the theoretical part;
  • describe standard components in FEM algorithms.

Course Disposition

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Literature and preparations

Specific prerequisites *

Single course students: 90 university credits including 45 university credits in Mathematics
or Information Technology. English B, or equivalent.

Recommended prerequisites

DN2221 Applied Numerical Methods, part 1 (or corresponding), can be read in parallel.

Equipment

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Literature

To be announced at least 4 weeks before course start at course web page. Previous year:
K. Eriksson, D. Estep, P. Hansbo, C. Johnson: Computational Differential Equations.
Studentlitteratur, ISBN 91-44-49311-8

Examination and completion

Grading scale *

A, B, C, D, E, FX, F

Examination *

  • LAB2 - Laboratory Work, 4.5 credits, Grading scale: P, F
  • TEN2 - Written Examination, 3.0 credits, Grading scale: A, B, C, D, E, FX, F

Based on recommendation from KTH’s coordinator for disabilities, the examiner will decide how to adapt an examination for students with documented disability.

The examiner may apply another examination format when re-examining individual students.

  • LAB2 - Laboratory Task, 4.5 credits, grade scale: P, F
  • TEN2 - Examination, 3.0 credits, grade scale: A, B, C, D, E, FX, F

Other requirements for final grade *

  • Examination (TEN2; 3 university credits).
  • Assignments (LAB2; 4.5 university credits).

Opportunity to complete the requirements via supplementary examination

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Opportunity to raise an approved grade via renewed examination

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Examiner

Sara Zahedi

Further information

Course web

Further information about the course can be found on the Course web at the link below. Information on the Course web will later be moved to this site.

Course web SF2561

Offered by

SCI/Mathematics

Main field of study *

Mathematics, Technology

Education cycle *

Second cycle

Add-on studies

Please discuss with the course leader.

Contact

Sara Zahedi (sara.zahedi@math.kth.se)

Ethical approach *

  • All members of a group are responsible for the group's work.
  • In any assessment, every student shall honestly disclose any help received and sources used.
  • In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.