Mathematical Foundations

A foundational course on differential equations, numerical methods and solver concepts behind multiphysics simulation.

ECTS credits

5.0 ECTS

Teaching hours

15 hours

Instructor

José Manuel González Vida

Department of Applied Mathematics

University of Málaga

This course is taught as a module in the Master’s Degree and can also be included in a custom program.

Course overview

In this course we review the mathematical foundations of multiphysics simulations: the general theory of partial differential equations and numerical methods for solving them, as well as other useful tools.

First, the general theory of Ordinary Differential Equations (ODEs) is introduced, followed by the theory of Partial Differential Equations (PDEs) using classical equations as examples.

Numerical methods for solving PDEs are then described, with the Finite Element Method (FEM) as the main tool. Direct and iterative solvers for the matrix algebraic equation are discussed for stationary and time-dependent cases.

The course also includes mathematical tools useful for solving algebraic systems, such as multigrid methods, domain decomposition methods and preconditioners.

Mathematical tools covered

Ordinary Differential Equations (ODEs).
Partial Differential Equations (PDEs).
Finite Differences.
Weak formulation.
Finite Element Method (FEM).
Boundary Element Method (BEM).
Direct and iterative solvers.
Preconditioners.
Time-dependent solvers.
Eigenvalue problem solvers.
Multigrid methods.
Domain decomposition.

Syllabus

Module 01

Ordinary Differential Equations (ODEs)

Initial value problems.
Boundary problems.
Scalar and vector cases with simple examples.
Numerical approximation.

Module 02

Partial Differential Equations (PDEs)

Fundamentals and classification.
Initial and boundary conditions.
Classical equations.
Fourier transforms.

Module 03

Numerical methods for PDEs

Introduction and motivation: Finite Differences.
Weak formulation.
The finite element method (FEM).
The boundary element method (BEM).

Module 04

Numerical solvers and COMSOL Multiphysics

Stationary problem solvers: direct and iterative.
Preconditioners.
Time-dependent problem solvers: implicit and explicit.
Eigenvalue problem solvers.
Multigrid methods and domain decomposition.

Understand the numerical foundations behind simulation.

Continue reviewing the course catalog or visit the teaching team behind the program.