Mathematical Foundations

Fall term

In this course we will 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. The various direct and iterative solvers that allow the solution of the matrix algebraic equation of the system, both in the stationary and time-dependent cases, are then discussed. The description will include other mathematical tools useful for solving the algebraic system, such as multigrid methods, domain decomposition methods and preconditioners, to name a few.

The content covers the mathematical foundations necessary and essential for you to understand the algorithms underlying numerical modeling in science and engineering. You will learn the basic operation of the solvers and be able to select and configure them according to the peculiarities and conditions of the problem to be solved.

- Initial value problems.
- Boundary problems.
- Scalar case and vector case with simple examples.
- Numerical approximation of their solutions.

- The partial differential equations (PDEs). Classification.
- Initial and boundary conditions.
- Classical equations.
- Fourier transforms.

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

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

Image made using the COMSOL Multiphysics® software and provided by courtesy of COMSOL.