Courses

Last Updated:
10/03/2019 - 15:02

IAM572 - Finite Element Methods for Partial Differential Equations: Theory and Applications

Credit: 3(3-0); ECTS: 8.0
Instructor(s): Ömür Uğur
Prerequisites: Consent of Instructor (Elementary knowledge of PDEs, basics of linear algebra, basic MATLAB coding skills)

Course Catalogue Description

Abstract Finite Element Analysis: weak derivatives, Sobolev spaces, Lax-Milgram lemma; Piecewise Polynomials Approximations 1D and 2D: interpolation, projection; Finite Element Method 1D and 2D: weak formulation, derivation of linear system of equations, a priori estimates; Time Dependent Problems: finite differences for systems of ODE, stability estimates; Semi-elliptic equations; a posteriori Error Analysis: estimator, mesh Refinement

Course Objectives

This course is designed for graduate students majoring in mathematics as well as mathematically inclined graduate engineering students. At the end of this course, the student will:

• learn the fundamental concepts of the theory of the finite element method
• learn how to formulate and solve second order partial differential equations in one and two spatial dimensions using finite element method
• learn how to solve time dependent non(linear) problems using finite differences in time and finite element method in space
• develop proficiency in the applications of the finite element method (modelling, analysis, and interpretation of results) to realistic engineering problems through the use of major commercial general purpose finite element code
• learn to implement finite element method using MATLAB.

Course Learning Outcomes

Upon successful completion of this course, the student will be able to:

• formulate and solve (with a computer) second order partial differential equations in one and two spatial dimensions using finite element method
• derive a priori error bounds for elliptic equations in one and two spatial dimensions
• solve time dependent partial differential equations using finite element method in space and finite differences in time, and to compare different time stepping algorithms and choose appropriate algorithms
• apply the finite element methods in their thesis and understand the current research in this area
• evaluate different techniques for solving problems and be able to motivate when to use existing software and when to write new code
• use MATLAB in their own project work.

Tentative (Weekly) Outline

1. Abstract Finite Element Analysis: weak derivatives, Sobolev spaces
2. Abstract Finite Element Analysis: Lax-Milgram lemma
3. Piecewise Polynomials Approximations 1D: interpolation, projection
4. Finite Element Method 1D: weak Formulation, derivation of linear system of equations
5. Finite Element Method 1D: computer Implementation
6. Finite Element Method 1D: a priori error estimates
7. Piecewise Polynomials Approximations 2D: meshes, interpolation, projection
8. Finite Element Method 2D: weak formulation, derivation of linear system of equations
9. Finite Element Method 2D: computer Implementation
10. Finite Element Method 2D: a priori error estimates
11. Time Dependent Problems: finite differences for systems of ODE
12. Time Dependent Problems: stability Estimates, computer Implementation
13. Semi-elliptic equations: discrete formulation
14. A Posteriori Error Analysis: estimator, mesh refinement

Course Textbook(s)

• M. G. Larson and F. Bengzon, The Finite Element Method: Theory, Implementation, and Applications, Springer-Verlag Berlin Heidelberg, 2013.

Supplementary Materials and Resources

• Books:
• M. S. Gockenbach, Understanding and Implementing the Finite Element Method, SIAM, 2006.
• Resources:
• MATLAB