Motivation:
Modern physics and technology problems often require solution of ordinary or partial differential equations (ODE or PDE). It is rare occasion that the solutions can be expressed analytically. Hence, researchers have to have a good command in numerical methods to obtain the result. One class of techniques to solve PDEs is the Finite Element Methods (FEM). There are some very modern commercial and open-source solvers that are based on these methods (e.g., Comsol). Therefore it is beneficial to gain some knowledge in the method theory as well as personal experience in its program implementation, to be able to professionally utilize broad capabilities of other available tools.
Expected course outcome:
Understanding of FEM theory basics, as well as ability to formulate computational algorithm and implement it as a Python program. Knowledge of specific differences between the FEM and the more publically known Finite Difference methods. Experience in solving practical boundary-value problems based on some 2nd order PDEs using the FEM. Understanding of the mesh generation aspect. The course will imply numerous practical assignments.
- Simple linear two-point boundary value problems in 1D. Overview of some alternative numerical solution techniques.
- Refreshment on Scientific Python programming. Mesh generation options in Python.
- Theoretical aspects of the FEM in 1D. Weak problem formulation. Approximation of various types of the boundary conditions. Matrix assembly technique. Account for the essential boundary conditions.
- FEM theory for the solution of 2D problems. Convergence analysis basics.
- Solution of the linear algebraic equations resulting from the FEM approximation. Sparse nature of the system and its account. Matrix storage techniques. Sparse matrix system solvers, their implementations in Python SciPy.
- Solution of transient problems.