Nature is by far the best Engineer worldwide and the human body is the living evidence of this statement: the dynamic interplay between skeleton, thought, breath, motion, muscles, and cardiovascular flow, is the driving force to everyday's life and a continuous inspiration for the design and development of novel materials, structures and hydraulic devices. Leveraging on this unexpected connection between Life Sciences and Civil Engineering, the course of Numerical Analysis aims at providing the student the basic instruments to construct a mathematical model of a realistic problem and the theoretical tools to develop and analyze methods and algorithms for its stable and accurate solution on a computer machine. The following topics will be treated during the course and numerically implemented using the Matlab scientific environment:

- approximation of functions and data, numerical integration;
- numerical solution of ordinary differential equations;
- numerical solution of linear algebraic systems;
- numerical solution of nonlinear equations and systems;
- numerical approximation and solution of partial differential equations with the Galerkin Finite Element Method.

The theoretical concepts, methods and algorithms will be numerically verified during the laboratory sessions through the solution of exercises and the critical discussion of project assignments. Exercise solution and project discussion will be conducted with the interactive participation (through remote web connection) of Prof. Giovanna Guidoboni, University of Missouri, Columbia MO, USA, a worldwide expert in the development and analysis of fluid-mechanical computational models in Ophthalmology and Fluid Flow in the Cardiovascular and Lymphatic systems. Projects will be subject to evaluation and will contribute to determine the final grade of the student.

The close interplay between class lectures and laboratories, supplied by the active participation of a worldwide expert such as Prof. Guidoboni, is expected to confer a "realistic" flavour to the course which is, oftentimes, rare in a mathematically-oriented context. Public discussion and correction of project assignments is also expected to significantly increase student active interaction with the theoretical subjects and their computer implementation. This latter outcome has a particular importance because it can be profitably used in other situations of students' life, such as thesis development and professional career. More in general, building upon the various components of this course (classes, laboratories, assignments, final exam), we foster students to be able to combine in a logically connected framework the basic elements of mathematical modeling, namely:

1. the description of the real problem;
2. the underlying assumptions that lead to the mathematical model;
3. the numerical techniques for model approximation;
4. the simulation tests;
5. the critical analysis of the results.

Being able to provide the above skills is the ultimate goal and expected outcome of a modern course of Numerical Analysis for students in Civil and Environmental Engineering.

- Foundations of numerical mathematics: the physical, mathematical and numerical problems. Existence, uniqueness and stability. Consistency; discretization error; convergence. Absolute and relative errors. The equivalence theorem. Sources of error in a computational process; floating point representation of real numbers; roundoff unit; machine epsilon. Example: numerical computation of the derivative of a function.

- Approximation of functions and data: polynomial interpolation, error analysis. Lack of convergence: Runge's counterexample. Piecewise polynomial interpolation, error analysis. Data fitting: least-squares approximation of clouds of data. Numerical integration of a function over a bounded interval: midpoint, trapezoidal, Cavalieri-Simpson and Gauss-Legendre composite quadrature rules; degree of exactness and order of accuracy of a quadrature rule.

- Numerical solution of systems of linear equations: Cramer's rule and its computational inefficacy. Alternate approaches: direct methods and iterative methods. Direct methods: the LU factorization, existence and uniqueness, pivoting. Stability analysis: the condition number of a matrix. Iterative methods: one-step linear iterative methods, iteration matrix, convergence, asymptotic convergence rate. The Richardson method. Examples: Jacobi, Gauss-Seidel and gradient methods. Stopping criteria and solution reliability.

- Numerical solution of ordinary differential equations (ODEs) and systems: the Cauchy problem: existence, uniqueness and stability of the solution. Forward and Backward Euler methods, the Crank-Nicolson method. Generalization: the theta-method. Consistency, stability and convergence. Absolute stability. The case of systems of ODEs. The use of Matlab odesuite package.

- Numerical solution of nonlinear equations and systems: fixed-point iterations; convergence analysis; order of convergence; stopping criteria and solution reliability. The Newton method. The case of system of nonlinear equations.

- Numerical approximation of balance laws in one spatial dimension: the time-dependent diffusion-advection-reaction model problem. Time semidiscretization with the theta-method. The Galerkin Finite Element Method: weak formulation, piecewise linear continuous finite elements, algebraic formulation, convergence, examples.

Knowledge of Calculus and Linear Algebra are required, as well as a familiarity with the basic principles of Physics and elementary Functional Analysis.

Student evaluation is the result of two contributions, the sum of which yields the final grade:

1. The first contribution is provided by the evaluation of a number of project assignments that must be elaborated by each student during the laboratories. Each project assignment will have a minimum grade of 0/30 and a maximum grade of 2/30, for a total of 10/30.

2. The second contribution is provided by the evaluation of a written test. Five exam sessions that will be scheduled according to the dates established by the calendar of the School of Civil and Environmental Engineering. The written test consists of the solution of a number of exercises organized according to the following structure:

• theoretical questions on the topics covered during the course, including: statement and/or proof of a theorem or mathematical property of a method; definition of concepts such as convergence, consistency, stability of a numerical scheme; formulation and analysis of a numerical method;
• computational questions to be numerically solved through the use of built-in MATLAB functions and MATLAB functions implemented during the laboratories, including: systems of linear equations, interpolation of data and functions, nonlinear equations, numerical integration and differentiation, approximation of ordinary and partial differential equations.

The written test will have a minimum grade of 0/30 and a maximum grade of 24/30. The evaluation process will take into account the correctness and the accuracy of the provided answers, the level of critical analysis of the obtained results, the ability in the use of MATLAB functions to solve basic mathematical problems as well as problems of interest in the engineering practice.

The theoretical content of the course and the numerical algorithms implemented in the Matlab scientific environment can be found in Chapters 3, 23, 28 and 29.