Luciano Teresi
Introduction to the Finite Element Method
6 CFU
Goal
The purpose of this course is to give an easily accessible introduction to the finite
element method as a general method for the numerical solution of PDE in
mechanics and physics, covering all the three main types of equations: elliptic,
parabolic and hyperbolic. The course will cover the following topics:
1. Introduction to FEM for elliptic problems.
Integral VS local formulations. The continuous problem and its discretization.
Hilbert spaces. The energy norm. Polynomial interpolation and error estimates.
Some finite element spaces. Regularity requirements; examples of finite
elements. Adaptive methods. Selected examples: the Poisson equation, the
elasticity problem; the Stokes problem.
2. Direct methods for solving linear systems of equations.
Gaussian elimination. Cholesky's method. Operation counts. Band matrices. Fillin. The frontal method. Nested dissection.
3. Minimization algorithms.
Iterative methods. The gradient method. The conjugate gradient method.
Preconditioning. Multigrid methods. Direct methods vs iterative methods.
4. FEM for parabolic problems.
Semi-discretization in space. Discretization in space and time. The backward
Euler and Crank-Nicolson methods. The discontinuous Galerkin method.
5. Hyperbolic problems.
A convection-diffusion problem. Standard Galerkin. Stabilization techniques:
classical artificial diffusion; streamline diffusion. The discontinuous Galerkin
method. The streamline diffusion method for time-dependent convection-diffusion
problems. Second order hyperbolic problems.
6. Boundary element methods.
Some integral equations. Exterior problems: Dirichlet – single and double layer
potential; Neumann - single layer potential.
7. Some examples of Mixed finite element methods
8. Selected non-linear problems. Convex minimization problems. A non-linear
parabolic problem. The incompressible Euler equations. The incompressible
Navier-Stokes equations. Compressible flow: Burgers' equation
Suggested reading
Claes Johnson, Numerical Solution of Partial Differential Equations by the Finite
Element Method, Cambridge University Press (1987).
Mark S. Gockenbach, Understanding and Implementing the Finite Element
Method, Society for Industrial and Applied Mathematics (2006).