**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.