The goal of the course is twofold. First, we present some classical models based on partial differential equations relevant to the context of continuum mechanics (fluid mechanics and solid mechanics), developing and analyzing analytical methods for the computation of the solutions and some finite difference schemes for their approximation. Second, we introduce the variational formulation of some boundary value problems, together with the finite element method for their numerical approximation. The course is characterized by a constant synergy between modeling, theoretical aspects and numerical simulation.

The lectures and the laboratories will provide students with:

i) knowledge and understanding (DD1) of
* the main properties of the solutions of some classes of linear partial differential equations;
* the basic tools used for solving linear partial differential equations;
* modeling of some physical problems
* finite difference approximation of time and space operators (of first and second order);
* finite element approximation in space (of first and seond order operators);
* the fundamental concepts of: consistency, stability and convergence;

ii) the ability to apply the previous knowledge (DD2) to simple examples. In particular the student is required to
* solve linear partial differential equations analytically and numerically
* discuss the properties of mathematical models
* be able to run and suitably modify a computer script provided by the instructor;
* critically discuss the resuls of the numerical experiments above in perspective of the theory.

The instructors expect a broad comprehension of the subjects which should not be limited to the statement of theoretical results.
Instead, the acquired knowledge should enable the students to express critical judgment (DD3) and make informed choices on analytical and numerical methods for partial differential equations. The students are expected to express their answers in a mathematically rigorous and clear way.

First Part – Differential modeling and finite difference approximation of partial differential equations

1. Review

Mathematical Methods: Differential calculus for functions of several real variables. Series of functions.

Numerical Methods: Finite difference formulae to approximate derivatives. Numerical approximation of ordinary differential equations, convergence, absolute stability.

1. First-order conservation laws

Mathematical Methods: Transport equation. Traffic flow models. Method of characteristics. Rankine-Hugoniot relation. Shock and rarefaction waves. Entropy condition.

Numerical Methods: Approximation with finite differences. Convergence, consistency, zero-stability and absolute stability. Forward Euler-centered scheme. Upwind,Lax-Friedrichs and Lax-Wendroff schemes. Analysis of the schemes, CFL condition and its meaning. Backward Euler-centered scheme. A quick description of systems and of non-linear problems.

2. Diffusion

Mathematical Methods: Heat equation. Well-posed problems. Separation of variables. Maximum principles. Fundamental solution. Cauchy problem in a strip and in the half-space with several boundary conditions (isolation, transmission, mixed). Duhamel principle.

Numerical Methods: Discretization of the heat equation with finite differences. Implicit and explicit time marching schemes, the theta-method, stability analysis.

3. Laplace-Poisson equation

Mathematical Methods: Harmonic functions. Mean value properties. Maximum principles. Well-posed problems. Poisson’s formula for the disk. Newtonian potentials.

Numerical Methods: Discretization with finite differences of a one-dimensional elliptic problem. Imposition of the Dirichlet and Neumann boundary conditions. Algebraic formulation and matrix properties. Diffusion-convection and diffusion-reaction problems.

4. Wave equation

Mathematical Methods: String equations. Well-posed problems and separation of variables. D’Alembert formula.

Numerical Methods: Discretization of the wave equation with finite difference explicit and implicit schemes. Leapfrog and Newmark schemes. Stability properties.

Second Part – Functional Analysis, variational formulations and discretizations via finite element method.

5. Basics of Functional Analysis

Mathematical Methods: Lebesgue integral. Projection theorem and Riesz representation theorem. Schwartz distributions. Sobolev spaces.

6. Weak formulation and Finite Elements approximation of stationary problems

Numerical Methods: Bilinear form, abstract variational problems and Lax-Milgram lemma. Variational formulation of elliptic problems and applications to transport-reaction-diffusion equations. Introduction to the Galerkin method for a one-dimensional elliptic problem. Consistency, stability and convergence. Cea' Lemma. The finite elements method. Linear and quadratic finite elements. Definition of Lagrangian basis functions, of composite interpolation and error estimates. Extension to the 2D case. Approximation of the diffusion-convection-reaction problem: comparison with the finite difference case and stability analysis. Stabilization with the upwind strategy and the mass lumping technique.

7. Evolution problems

Numerical Methods: Approximation with the Galerkin method, the semi-discrete problem. Explicit and implicit time marching schemes, the theta-method. Stability properties. A quick description of finite elements for hyperbolic problems.

• SDG9 - INDUSTRY, INNOVATION AND INFRASTRUCTURE

We recommend that students who attend this course have knowledge of linear algebra, calculus and numerical analysis, in particular:

* differential calculus for functions of several real variables;
* surfaces and integral surfaces, the divergence theorem;
* series of functions, Fourier series;
* linear second order ordinary differential equations;
* numerical solution of linear systems (by direct and iterative methods);
* polynomial interpolation;
* basic numerical methods for the approximation of ordinary differential equations;

There are five examination dates (two in January-February, two in June-July, one in September). The course consists in two moduli, one in Mathematical Methods (ref. F. Giuliani and F. Punzo) and one in Numerical Methods (ref. C. Vergara and P. Zunino).
The exam in Mathematical Methods is written and consists in two questions on the theory (6 points each) and two exercises (10 points each). A positive evaluation requires at least 6 points on the theory and 10 points on the exercises, with the total grade being at least 18.
The exam in Numerical Methods consists in a written part and an optional oral part. Students can take the oral exam for the Numerical Methods modulus only when the corresponding written grade is at least 15 out of 30. A positive evaluation requires a total grade of at least 18. It is mandatory to take the written and the oral part of the Numerical Methods modulus within the same examination date, i.e. it is not possible to give the written part in one examination date and the oral part in another one.

The final grade is the (rounded up) arithmetic mean of the grades obtained in the two moduli. To get the grade “30 cum laude” one should obtain at least 32 in average.

The written part of both moduli takes place in the same day. It is possible to take the exam in just one modulus (Mathematical or Numerical Methods) in one of the five examination dates and in the other modulus in another examination date, provided such dates are in the same academic year.

The participation to the written exam of one of the two moduli, automatically discards any previous grade obtained for that modulus, even if the student chooses to withdraw.

For the numerical part the exam will be done by using Microsoft form in classroom (unless there are restrictions due to COVID-19 or other), so the day of the written exam the students must have a labtop with a functioning Polimi wireless connection. The written exam consists in three parts: open questions, Matlab or FreeFem exercise, questions about theory.

NOTES

1)  Mathematical Methods in Engineering and Numerical Methods in Engineering are also single courses which can be taken independently.

2)  Browsing texts, notes, and electronic devices are not allowed during the tests. It is mandatory to bring an ID (e.g. identity card, driver’s licence,…) in order to be identified.

3)  Registration to the exam is mandatory, before the deadline. Lately registered students will not be admitted, no exceptions will be made.

The exam is designed to test the student on the following skills:

i) the knowledge and understanding of

* the main properties of the solutions of some classes of linear partial differential equations;
* the basic tools used for solving linear partial differential equations;
* modeling of some physical problems
* finite difference approximation;
* finite element approximation in space;
* the ability to solve simple exercises based on all the metods that have been introduced in the course;
* the concepts of consistency, stability and convergence applied to all the metods that have been introduced in the course;

ii) the ability to apply the previous knowledge to simple examples. In particular the student is required to

* solve exercises on linear partial differential equations;
* discuss the properties of mathematical models;
* be able to run and suitably modify a computer script provided by the instructor;
* critically discuss the resuls of the numerical experiments above in prespective of the theory;

The instructors expect a broad comprehension of the subjects which should not be limited to the statement of theoretical results.

Instead, the acquired knowledge should enable students to express critical judgment and make informed choices on analytical and numerical methods for partial differential equations.

The students are expected to express their answers in a mathematically rigorous and clear way.