Numerical Methods for Partial Differential Equations

Recommended Prerequisites

Differential and Integral Calculus, Linear Algebra, Numerical Analysis and Programming.

Teaching Methods

Classes are expository and include examples and exercises for applying the acquired knowledge. As homework the students solve analytical or computational problems that involve the application of the methods studied.
During the semester students may use tutorial time to clarify their difficulties in grasping the theory and in gaining practical knowledge, as well as in the development of the necessary skills for the computational assignment.

Learning Outcomes

The aim of this course is to develop skills to solve numerically steady and evolution partial differential problems and to analyze and interpret the computed solutions. Particularly, the course aims to endow the students with the theoretical and practical foundations   of finite difference methods, Galerkin methods   and   numerical methods for conservation laws.

Work Placement(s)

No

Syllabus

I. Steady problems:

Finite difference methods – stability and convergence;

Galerkin methods – variational formulation and Céa’s lemma;

Finite element methods- finite element spaces, error estimates.

II. Evolution problems:

Method of Lines (finite difference methods  and Galerkin methods) -  stability and convergence.

III. Numerical methods for conservation laws.

Head Lecturer(s)

Adérito Luís Martins Araújo

Assessment Methods

Continuous assessment
During the semester there a final exam (60%-80% of the final grade) and a set of analytical or computational homework assignments (40%-20% of the final grade.: 100.0%

Final assessment
The final exam option consists of a single exam (100% of the final grade).: 100.0%

Bibliography

W. Hackbush, Elliptic Differential Equations: Theory and Numerical Treatment, Springer, 1987.

J.W. Thomas, Numerical Partial Differential Equations: Finite Difference Methods, Springer, 1995.

S.C. Brenner, L.R. Scott, The Mathematical Theory of Finite Element Methods, Springer, 1991.

G. Sod, Numerical Methods in Fluid Dynamics: Initial and Initial Boundary Value Problems, Cambridge University Press, 1988.

V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Lectures Notes in Mathematics, Vol. 1054, Springer, 1984.

R.J. Leveque, Numerical Methods for Conservation Laws, Birkhauser, 1992.