NUMERICAL APPLICATIONS IN MATERIALS SCIENCE
Learning outcomes of the course unit
The main aim of this course is to provide a thorough illustration of numerical methods, especially those stemming from the formulation of PDEs, carry out their stability and convergence analysis, derive error bounds and discuss the algorithm aspects relative to their implementation.
Contents of course: METODI NUMERICI PER LE APPLICAZIONI o ELEMENTI DI ANALISI NUMERICA.
Course contents summary
Numerical solution of linear systems. Classical iterative methods: Jacobi method, Gauss-Seidel method, relaxation methods. Modern Iterative methods:preconditioned Richardson method, Conjugate Gradient method, Conjugate Gradient for non-symmetric problems.
Elliptic Problems: approximation by Galerkin and collocation methods. Variational form of boundary value problems. Existence, uniqueness and regularity of solutions. Galerkin method: finite element and spectral approximations. Orthogonal polynomials. Gaussian quadrature and interpolation. Generalized Galerkin method.
Steady Advection-Diffusion problems. Weak formulation. A one-dimensional example. Galerkin approximation and centered finite differences. Upwind finite differences and numerical diffusion.
Parabolic problems. Mathematical analysis and initial-boundary value problems. Semi-discrete approximation. Time-advancing by finite differences.
Hyperbolic problems. Some instances of hyperbolic equations. Linear scalar advection equations. Linear Hyperbolic systems. Approximation by finite differences. Stability, consistency and convergence. Approximation by finite elements. Galerkin method. Space-discontinuous Galerkin method. Schemes for time-discretization.
A.Quarteroni, R.Sacco, F.Saleri, Matematica Numerica, Springer-Verlag.
A. Quarteroni, A. Valli, Numerical Approximation of Partial Differential Equations, Springer-Verlag.