The aim of this course is twofold: on the one hand, to extend the one-dimensional finite element analysis for partial differential equations (PDEs) to high dimensions, with an increased attention to applicative problems in Science and Engineering; on the other hand, to enrich the theoretical analysis with laboratory sessions to provide the students with a practical counterpart of what stated by the theory. Scientific talks given by expert scientists and dealing with some of the latest challenging topics in scientific modeling are also planned during the course.
+ Multidimensional elliptic problems: the 2D Poisson problem; general elliptic problems; the Lax-Milgram lemma; Galerkin approximation; Galerkin orthogonality; stability and convergence analysis; the finite element method; interpolation error estimates; a priori and a posteriori error estimators; mesh adaptivity; dual problems.
+ Multidimensional advection-diffusion problems: limits of the Galerkin method for a convection dominated problem; stabilized finite element schemes; the generalized Galerkin formulation; the artificial diffusion and the upwind schemes; strongly consistent stabilized methods; stability and convergence analysis for the GLS scheme; stabilization based on bubble functions; the mass lumping technique for reaction dominated problems.
+ Multidimensional time dependent problems: the semidiscrete formulation; the theta-method; a priori estimates; convergence analysis for the semidiscrete form; stability analysis for the fully discrete formulation; convergence result for the fully discrete problem.
+ The one-dimensional Galerkin spectral method: Legendre polynomials; Gauss-Legendre-Lobatto quadrature rules; the G-NI method; equivalence between the G-NI and a collocation scheme; the Strang Lemma; convergence of the G-NI formulation; extension to a two-dimensional setting; the spectral element method (SEM); the SEM-NI approach.
+ One-dimensional hyperbolic problems: scalar linear equations; a priori analysis; finite element approximation in space; time discretization with the implicit and explicit Euler schemes; stability analysis; time discretization with the LF, LW, UPWIND schemes; strong stability analysis; dissipation and dispersion analysis; the Taylor-Galerkin schemes.
+ Two-dimensional hyperbolic problems: inflow and outflow boundaries; strong and weak imposition of the boundary conditions; stability and convergence analysis; time discretization with the implicit and the explicit Euler schemes; stability analysis; the discontinuous Galerkin (DG) method.
Labs and exercise sessions:
Most of the methods presented during the lectures will be numerically investigated during the laboratory sessions. Sessions of supervised exercises are also provided in view of the written test.
Analisi Numerica delle Equazioni alle Derivate Parziali I.