Politecnico di Torino
Academic Year 2017/18
02JNZNG
Numerical methods for partial differential equations
Master of science-level of the Bologna process in Mathematical Engineering - Torino
 Teacher Status SSD Les Ex Lab Tut Years teaching Canuto Claudio PO MAT/08 50 30 0 0 15
 SSD CFU Activities Area context MAT/08 8 B - Caratterizzanti Discipline matematiche, fisiche e informatiche
 Subject fundamentals The course is an introduction to some general methodologies for the numerical treatment of partial differential equations modelling phenomena of engineering interest. Expected learning outcomes A particular emphasis will be given to finite element methods, spectral methods and finite volume methods. In addition, iterative methods will be described to solve high dimensional algebraic systems (both linear and nonlinear) Prerequisites / Assumed knowledge Numerical Methods Contents Boundary value problems for partial differential equations Galerkin and Petrov-Galerkin methods; link with collocation methods. Approximation of functions through piecewise polynomial functions; finite elements and their approximation properties. Finite element methods; stability, consistence, and convergence; a priori and a posteriori error estimates; grid generation and adaptivity; algebraic systems deriving from finite element discretization. Spectral methods and their properties. Fourier, Chebishev, and Legendre methods. Spectral element methods. Conservation laws and finite volume methods. Consistence and stability. Courant number and CFL condition. Numerical diffusion and dispersion. Iterative methods for high dimensional algebraic systems; preconditioning; gradient methods and Lanczos-like methods; multilevel methods. Examples. Delivery modes Exercises will be aimed at training the student in the use of mathematical software for the solution initial-boundary value problems for partial differential equations. Texts, readings, handouts and other learning resources P.A: Raviart e J.M. Thomas, Introduzione all'analisi numerica delle equazioni alle derivate parziali, Masson, Milano 1988. A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer, Berlin 1997. A. Greenbaum, Iterative methods for solving linear systems, SIAM, Philadelphia, 1997. Programma definitivo per l'A.A.2014/15

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