The course aims at providing an introduction to the concepts of numerical approximation, error analysis and computational methods that represent the main constituents of Numerical Mathematics. After providing motivating example to this study, such as the dynamical distribution of temperature and heat flux in a bar under the action of externally applied forces and of distributed sources/sinks of heat, basic concepts and methods of Numerical Analysis will be introduced, analyzed and implemented on the computer. Then, these methodologies will be applied in the context of the Galerkin Finite Element Method (GFEM) to numerically approximate elliptic and parabolic partial differential equations (PDEs) in one spatial dimension (1D).

The course is organized in lectures, exercises and computer labs. Exercise and laboratory sessions are meant to provide a practical feedback to the theoretical knowledge with the implementation of the algorithms and methods presented during the lectures. All computations are carried out using the Matlab scientific software environment.

The lectures, exercises and computer labs will allow the students, who have successfully passed the exam, to know and understand:­

‐ basic concepts of Numerical Analysis with particular regard to the approximation methods to solve linear systems, approximation of data and functions, approximation of integrals;

- the main properties and the well-posedness of the Cauchy problem for ordinary differential equations;

- numerical methods and algorithms to approximate ordinary differential equations (ODEs);

- the stability and convergence analysis of the schemes to approximate ODEs;

- the variational formulation of 1D elliptic and parabolic boundary value problems;

- numerical methods and algorithms for the approximation of 1D elliptic and parabolic boundary value problems, in particular via a Finite Element (FE) discretization;

- error estimates to certify the accuracy of the FE discretization of 1D elliptic and parabolic boundary value problems;

- the use of MATLAB software for the implementation of scripts and functions to code the numerical methods considered.

 Thus, the acquired knowledge can be employed for:

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- the implementation of numerical algorithms for solving common problems in engineering practice that require the solution of linear systems, the approximation of data and functions, the integration of functions;

- the solution of mathematical problems formulated in terms of ODEs and PDEs by suitable numerical methods;

- critical reasoning and interpretation of the results obtained;

‐ the choice of the numerical method best suited to the solution of basic mathematical problems such as the solution of linear systems, the approximation of data and functions, the integration of functions;

­‐ the choice of the numerical method best suited to the solution of problems of applicative interest that can be modeled by elliptic or parabolic PDEs.

The student is expected to show a critical understanding of the contents of the course, without limiting to the presentation of definitions and results. Furthermore, students are expected to be able to solve the exercises with a rigorous and logical approach consistently with the theoretical knowledge.

A: Fundamentals of Numerical Analysis:

- Numerical methods for the approximation of functions and data: Lagrange form of the polynomial interpolation; piecewise interpolation; cubic interpolating splines; least-squares approximation of clouds of data;

- Numerical methods to approximate definite integrals: simple and composite formulas; midpoint, trapezoidal, Cavalieri-Simpson quadrature rules; Gaussian formulas; degree of exactness and order of accuracy of a quadrature rule;

- Numerical approximation of ODEs: one-step methods (forward and backward Euler and Crank-Nicolson schemes); consistency, zero-stability, convergence, absolute stability; systems of ODEs (hints);

- Numerical approximation of systems of linear equations: direct methods (Gaussian elimination method, LU factorization, pivoting, Cholesky factorization); iterative methods (stationary and dynamic Richardson schemes, stopping criteria); accuracy and stability of the approximation; the condition number of a matrix and the preconditioners;

B: Galerkin Finite Element Methods for PDEs:

- GFEM approximation of elliptic problems in 1D: advection-diffusion-reaction equations; Dirichlet, Neumann and Robin boundary conditions; Sobolev spaces (hints); variational formulation; well-posedness of the weak form; numerical approximation with the Galerkin formulation; the finite element discrete space; error estimates for the finite element discretization; algebraic formulation of a finite element discretization; properties of the stiffness matrix; some examples of interest for the Engineering;

- GFEM approximation of parabolic problems in 1D: discretization in space with the finite element scheme; time discretization with the theta-method; convergence and stability properties; example: modeling and simulation of the dynamical distribution of temperature and heat flux in a bar under the action of externally applied forces and of distributed sources/sinks of heat.

Knowledge of Mathematical Analysis, Calculus and Linear Algebra are required.

Five exam sessions are scheduled, on the dates established by the Faculty calendar.

The exam consists of a written test during which it will be required:

• to answer to theoretical questions on the topics covered during the course, which may deal with the statement and/or the proof of relevant theoretical results; the precise definition of concepts such as convergence, consistency, stability of a numerical scheme; the deduction of a numerical method;
• to use built-in MATLAB functions and MATLAB codes implemented during computer labs for the solution of numerical problems, including systems of linear equations, interpolation of data and functions, numerical integration, approximation of ODEs;
• to use MATLAB codes implemented during computer labs to solve 1D elliptic and parabolic PDEs in view of the modeling of practical problems of interest for Engineering.

The exam evaluation will take into account the correctness and the accuracy of the provided answers, the critical reasoning capacity, the ability to implement MATLAB algorithms and to use the MATLAB codes implemented during computer labs to solve basic mathematical problems as well as problems of interest in Engineering.