Numerical Methods for Partial Differential Equations

The notes are taken from the books required for the course:

📚 Course Syllabus

Variational formulation of PDEs.
- Elliptic and parabolic PDEs (Poisson, advection-diffusion-reaction, and heat equations);
- Boundary and initial conditions;
- Strong and weak formulations of the PDEs;
- Lax-Milgram lemma.
Finite Element method for elliptic PDEs in 1D/2D/3D.
- Galerkin method, consistency, stability and convergence.
- The finite element method in 1D/2D/3D;
- Continuous Lagrangian basis functions;
- Finite elements and meshes;
- Algebraic properties of the fully discrete problem;
- Accuracy and computational costs;
- Error estimates and error analysis.
- Finite element approximation of advection-reaction-diffusion equations and vectorial problems.
Numerical approximation of time-dependent problems.
- Heat equation, semi-discrete problem and Galerkin method.
- Time discretization, explicit and implicit methods, theta-method, accuracy and stability properties.
Advanced topics.
- Finite element approximation of saddle-point problems, Stokes equations.
- Finite Element approximation of nonlinear PDEs, Newton method.
- Multiphysics problems and PDEs coupled through interfaces.
- Domain decomposition methods and introduction to parallel computing and preconditioners.

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