NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS

Ph.D. Programme in Civil and Environmental Engineering, International cooperation and Mathematics (DICACIM)
XXXV Cycle (A.Y. 2019/20)

Programme

• Elliptic equations:
  • approximation via the Galerkin method;
  • the Finite Element Method;
  • the Spectral Element Method;
  • error estimates.
• Parabolic equations:
  • the semi-discrete problem, convergence analysis;
  • the theta-method for time discretization: stability and convergence analysis.
• Diffusion-transport-reaction equations:
  • stabilization methods.

MATLAB campus license

Textbooks

A. Quarteroni, Numerical Models for Differential Problems (free download for Unibs users).
A. Quarteroni, F. Saleri, P. Gervasio Calcolo Scientifico (free download for Unibs users).
A. Quarteroni, F. Saleri, P. Gervasio Scientific Computing.

Videos

The videos of both synchronous and asynchronous lessons can be found at the page of the e-learning community Numerical Methods for Partial Differential Equations.
The interested students are required to send me an e-mail (paola_dot_gervasio_at_unibs_dot_it) in order to register for the community.

Lesson 1
Lecture (1h)	Introduction to PDEs.	slide1.pdf
Lesson 2
Lecture (3h)	Elliptic problems and Elements of Functional Analysis. Strong form of 2nd order elliptic problems for d=1. The space L^2. Functionals and forms on normed spaces. Distributions and differentiation of distributions. The Sobolev space H^1. Weak form of the Poisson problem with homogeneous Dirichlet boundary conditions for d=1. Non-homogeneous Dirichlet conditions for d=1. 2nd-order elliptic problems with Neumann boundary conditions for d=1. Lax-Milgram lemma.	slide2.pdf
Lesson 3
Lecture (4h)	Finite Elements approximation of 1D elliptic problems. 3a. Galerkin formulation of the discrete problem. The algebraic formulation of the discrete problem. The Cea Lemma. Estimate of the continuity and of the coercivity constants for the bilinear form a(u,v)=(u',v')+sigma(u,v). 3b. Linear finite elements: the Lagrangian basis, construction of the mass and stiffness matrices. 3c. Construction of the right hand side. Gauss-Legendre quadrature formulas. Quadratic finite elements. The Lagrange composite interpolation (of degree 1 and of degree r gt 1). Estimate of the error between the continous solution of the pde and the discrete solution. 3d. The 2nd order elliptic problem with non-homogeneous Dirichlet conditions and with Neumann conditions. Connectivity matrix.	slide3_a.pdf slide3_b.pdf slide3_c.pdf slide3_d.pdf
Lesson 4
29/04/2020 h. 9.00- 12.00 Laboratory (3h)	MATLAB - FEM 1d for elliptic problems.	slide4.pdf FEM1d.zip Lab1_MATLAB.zip
Lesson 5
Lecture (2h)	Elliptic problems for d ≥ 2. Finite Elements discretization. Weak formulation of 2nd order elliptic problem. Triangulations. P1 and Q1 FEM. Matrix assembling. Quadrature formulas. Cholesky factorization to solve the linear system.	slide5_ab.pdf
Lesson 6
Lecture (1h)	A short review on Spectral Element Methods.	slide6_a.pdf
Lesson 7
Lecture (2h)	Approximation of Parabolic problems. Strong form of the parabolic equations (heat equation). Semidiscrete weak form and Galerkin approximation. Semidiscrete numerical solution. Approximation of the first order Cauchy problem by Euler and Crank-Nicolson methods. Theta-method for the discretization of the heat equation. Absolute stability and convergence of the theta-method applied to the heat equation.	slide7x.pdf
Lesson 8
08/05/2020 h. 15.00 - 17.00 Laboratory (2h)	MATLAB PDEtoolbox	slide8.pdf Lab2.zip
Lesson 9
15/05/2020 h. 9.00 - 11.00 Laboratory (2h)	MATLAB discretization of the heat equation.	slide9.pdf FEM_1d_heat.zip Lab3.zip
Lesson 10
Lecture (1h)	Advection Diffusion Reaction problems. Strong and weak formulation. Galerkin approximation. 1D case with linear FEM: the Peclet number and the bound on h to avoid numerical oscillations. Upwind scheme and centered scheme with artificial diffusion. 2D case: an overview on artificial diffusion, streamline diffusion and Galerkin Least Squares methods.	slide10x.pdf
Lesson 11
15/05/2020 h. 11.15-12.30 Laboratory (1h+ 15')	MATLAB advection diffusion problems.	slide11.pdf FEM_1d_ad.zip FEM_2d_ad.zip Lab4.zip

Paola Gervasio   -   May 2020