. – Elements of numerical analysis
Text under revision. Not yet approved by academic staff.
To teach students the basis they need to solve limit problems numerically
(discretization), with special reference to the finite element method for elliptic
partial differential equations. Related aspects include: solving linear systems with
modern iterative methods and function approximation techniques.
Approximations of one-variable functions: Hermite interpolation and spline
function interpolation. Optimal approximation.
Linear systems with sparse matrices: gradient method and conjugate gradient
method (case of symmetric positive definite matrices); methods for nonsymmetric
matrices (BCGstab and GMRES).
Preconditioners for linear systems. Overview of the multigrid method.
Limit problems in several dimensions: Galerkin method and finite elements,
interpolation error and energy-norm error estimates.
Elliptic equations (Poisson's equation): L2 error estimate.
Computational problems: grid generation and matrix assembly.
Adaptive methods for partial differential equations (overview).
Overview of the case of parabolic and hyperbolic equations.
Univariate optimization problems (overview).
V. COMINCIOLI, Analisi Numerica, Metodi Modelli Applicazioni, McGraw Hill, Libri Italia,
Milan, 1990.
A. QUARTERONI - A. VALLI, Numerical approximation of partial, differential equations,
Springer, 1994.
C. JOHNSON, Numerical solution of partial differential equations by the finite element method,
Cambridge university press, Cambridge, 1990.
G.H. GOLUB - C.F. VAN LOAN, Matrix Computations, The Johns Hopkins University Press,
Baltimore and London, 1993.
Further information can be found on the lecturer's webpage at
http://www2.unicatt.it/unicattolica/docenti/index.html or on the Faculty notice

Elements of numerical analysis

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