Numerics of Partial Differential Equations, University of Vienna, WS 2018
Schedule: Mon 8:00-9.30 SR09, Thu 8:00-9.30 HS02, Oskar-Morgenstern-Platz 1 (Information: u:find)
Course notes: see Moodle
Program:
01.10.2018: Course introduction; elements of functional spaces: spaces of continuous functions, spaces of integrable functions; distributions and distributional derivatives
04.10.2018: Sobolev spaces; Poincaré-Friedrichs inequality, some Sobolev embeddings, traces
08.10.2018: Variational formulation of elliptic problems; the Lax-Milgram lemma; Galerkin methods: definition, matrix form
11.10.2018: Galerkin methods: well-posedness, Céa's lemma and convergence; remarks on the symmetric case, on the stiffness matrix, and on advection-dominated advection-diffusion problems
22.10.2018: Generalized Galerkin methods: Strangs' lemma; introduction to Finite Element Methods (FEM): meshes
25.10.2018: Introduction to FEM: spaces of piecewise polynomial functions, basis functions and degrees of freedom
29.10.2018: Interpolation operators; FEM approximation of the Poisson problem
05.11.2018: FEM approximation of the Poisson problem: computing local matrices and rhs, assembling of global matrices and rhs, imposition of Dirichlet boundary conditions
08.11.2018: Matlab implementation of P1-FEM for the Poisson problem with Dirichlet boundary conditions
12.11.2018: Matlab implementation of P1-FEM for the Poisson problem with Dirichlet boundary conditions: tests and convergence plots
15.11.2018: Interpolation errors: scaling argument, Bramble-Hilbert's lemma
19.11.2018: Deny-Lions' lemma; bounds of the local and of the global interpolation errors
22.11.2018: Estimate of the FEM error in the H1 norm; estimate of the FEM error in the L2 norm: duality argument
26.11.2018: Complements: inverse estimate, bound of the condition number of the stiffness matrix; the Helmholtz problem: definition and variational formulation
29.11.2018: Derivation of the Helmholtz equation from the wave equation in the time-harmonic case; FEM approximation of the Hemlholtz problem
03.12.2018: Advection-diffusion problems in 1D: numerical instability in the advection-dominated case
06.12.2018: Advection-diffusion problems in 1D: artificial diffusion
10.12.2018: Advection-diffusion problems in 2D: streamline diffusion; strongly consistent stabilization method: SUPG, GALS, DWG; well-posedness of GALS
13.12.2018: Error analysis of GALS
07.01.2019: The heat equation; semidiscretization in space
10.01.2019: Matlab implementation of P1-FEM for advection-diffusion problems with artificial diffusion
14.01.2019: The heat equation: discretization in time and stability analysis of the theta-method
17.01.2019: Matlab implementation of P1-FEM for the Helmholtz problem
21.01.2019: Finite element library NGSolve (Part I)
24.01.2019: Finite element library NGSolve (Part II)
28.01.2019: Questions
31.01.2019: Exams (by appointment only)