Numerics of Partial Differential Equations, University of Vienna, WS 2019

Schedule: Mon 11:30-13:00 SR08, Fri 11:30-13:00 SR12, Oskar-Morgenstern-Platz 1 (Information: u:find)

Course notes: regularly posted in Moodle

04.10.2019: Course introduction; preliminaries: element of functions spaces (spaces of integrable functions, Hilbert spaces)
07.10.2019: Poincaré-Friedrichs inequality, some Sobolev embeddings, traces; linear, second order, elliptic problems
11.10.2019: Variational formulation of elliptic problems, the Lax-Milgram lemma
14.10.2019: Galerkin methods: definition, matrix form, well-posedness, Céa's lemma and convergence; remark on the symmetric case
18.10.2019: Remark on advection-dominated advection-diffusion problems; Generalized Galerkin methods: Strangs' lemma
21.10.2019: Introduction to Finite Element Methods (FEM): meshes, spaces of piecewise polynomial functions
25.10.2019: Introduction to FEM: basis functions and degrees of freedom; interpolation operators; FEM approximation of the Poisson problem
28.10.2019: FEM approximation of the Poisson problem: computing local matrices and rhs, assembling of global matrices and right-hand side, imposition of Dirichlet boundary conditions
04.11.2019: Matlab implementation of P1-FEM for the Poisson problem with Dirichlet boundary conditions
08.11.2019: Matlab implementation of P1-FEM for the Poisson problem with Dirichlet boundary conditions (conclusion)
11.11.2019: Interpolation errors: scaling argument, Bramble-Hilbert's lemma
15.11.2019: Deny-Lions' lemma; bounds of the local and of the global interpolation errors
18.11.2019: Estimate of the FEM error in the H1 norm; estimate of the FEM error in the L2 norm: duality argument
22.11.2019: Complements: an inverse estimate; the Helmholtz problem: derivation from the wave equation, variational formulation, continuity and coercivity of the sesquilinear form
25.11.2019: FEM approximation of the Hemlholtz problem: theoretical analysis, implementation, and numerical results
29.11.2019: Advection-diffusion problems in 1D: numerical instability in the advection-dominated case
02.12.2019: Advection-diffusion problems in 1D: artificial diffusion; 2D implementation and numerical results
06.12.2019: Advection-diffusion problems in 2D: artificial and streamline diffusion; strongly consistent stabilization method: SUPG, GALS, DWG; GALS: coercivity, existence and uniqueness of discrete solutions
09.12.2019: GALS: continuous dependence on the data, error analysis
13.12.2019: Conclusion of the error analysis of GALS; the heat equation: energy dissipation
10.01.2020: The heat equation: semidiscretization in space; discretization in time
13.01.2020: Numerical tests with NGSolve (with forward Euler and backward Euler); stability analysis of the theta method for 1/2 ≤ theta ≤ 1 (unconditional stability)
17.01.2020: Stability analysis of the theta method for 0 ≤ theta < 1/2 (conditional stability and CFL condition); convergence rates (no proof)
20.01.2020: Discontinous Galerkin (DG) discretization of advection-diffusion problems
24.01.2020: DG discretization of advection-diffusion problems: conclusion (no analysis)
27.01.2020: Conforming discretization spaces on polygonal meshes: virtual elements (k=1 only, no proofs)
31.01.2020: Final exams (by appointment to be taken within Sun, Jan 26)