Numerics of Partial Differential Equations, University of Vienna, WS 2020

Schedule: Mon 10:00-13:00 Online (Information: u:find)

Course notes: regularly posted in Moodle

Program:
05.10.2020: Preliminaries: elements of functions spaces (spaces of integrable functions, Hilbert spaces)
12.10.2020: The Poincaré-Friedrichs inequality, some Sobolev embeddings, traces; linear, second order, elliptic problems; variational formulation of elliptic problems, the Lax-Milgram lemma
19.10.2020: Galerkin methods: definition, matrix form, well-posedness, Céa's lemma and convergence; remark on the symmetric case
09.11.2020: Remark on advection-dominated advection-diffusion problems; Generalized Galerkin methods: Strangs' lemma
16.11.2020: Introduction to Finite Element Methods (FEM): meshes, spaces of piecewise polynomial functions, basis functions and degrees of freedom; interpolation operators
23.11.2020: FEM approximation of the Poisson problem: computing local matrices and rhs, assembling of global matrices and right-hand side, imposition of Dirichlet boundary conditions; Matlab implementation of P1-FEM for the Poisson problem with Dirichlet boundary conditions
30.11.2020: Interpolation errors: scaling argument, Bramble-Hilbert's lemma, Deny-Lions' lemma; bounds of the local and of the global interpolation errors
07.12.2020: Estimate of the FEM error in the H1 norm; estimate of the FEM error in the L2 norm: duality argument
14.12.2020: The Helmholtz problem: derivation from the wave equation, variational formulation, continuity and coercivity of the sesquilinear form; FEM approximation of the Hemlholtz problem: theoretical analysis
11.01.2021: Advection-diffusion problems in 1D: artificial diffusion; advection-diffusion problems in 2D: artificial and streamline diffusion
18.01.2021: Advection-diffusion problems in 2D: strongly consistent stabilization method: SUPG, GALS, DWG; GALS: coercivity, existence and uniqueness of discrete solutions, and continuous dependence on the data; error estimates (statement); numerical results on artificial diffusion.

Exam: Oral exam (online). Please make an appointment by email.