DN2260 The Finite Element Method 6.0 credits

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, convection-diffusion, Navier-Stokes equation, quantum mechanics etc.

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, for which the student should 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.

In the practical part of the course the student should be able to:

• modify an existing FEM program to solve a new scalar linear PDE.
• implement an adaptive mesh refinement algorithm, based on an a posteriori error estimate derived in the theoretical part.
• describe standard components in FEM algorithms.

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

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

To be announced at least 4 weeks before course start at course web page. Previous year: material produced at the department was used.

If the course is discontinued, students may request to be examined during the following two academic years.

• LAB2 - Laboratory Task, 3.0 credits, grading scale: P, F
• TEN2 - 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.

In this course all the regulations of the code of honor at the School of Computer science and Communication apply, see: http://www.kth.se/csc/student/hederskodex/1.17237?l=en_UK.

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

• 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.

2D1263 Scientific Computing, 2D1290/DN2290 Advanced Numerical Methods, or 2D1252/DN2252 or 2D1253/DN2253.