MTH 654/9: Finite Element Methods - Fall 2021
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General information
Class announcement
General information
Instructor: Malgorzata Peszynska, Professor of Mathematics (Contact information including office hours on instructor's department website)
Class: Lecture: MWF 15:00-15:50pm, STAG 111.
Course information: see CANVAS.
Schedule: {Assignments}
• 9/22: Introductions. What is FEM and why FEM and what this class is not about.
• 9/24: FE for 1d Poisson problem.
• 9/27: Inner product and normed spaces. Function norms. Numerical integration. {First day survey}
• 9/29: Code FDFEM1_singular.m. Weak (distributional) derivatives.
• 10/1: Examples of when $f \not \in C^0$. Assembly calculations, and reference element calculations (outlook towards $hp$ FE). Functionals and distributions: examples {HW1 due}
• 10/4: (M) equivalent to (V). Basic H_0^1 error estimate.
• 10/6: Poincare-Friedrichs inequality, and how to simplify basic error estimate. Bilinear forms.
• 10/8: Worksheet/handout and group work on bilinear forms {HW2 due 10/10/21}
• 10/11: HW2 discussion.
• 10/13: Solving more general BVP for more general PDEs with FEM. Neumann conditions; first and zero order terms.
• 10/15: Construction of FE code via element calculations including higher order elements, and assembly. MP code fem1d2021_forclass.m to be used for HW4 and HW3.
• 10/18: Aubin-Nitsche lemma and proof. Why using grid norms is a bad idea for FE calculations.
• 10/20: Group work on: which k to choose when $u \in H^r$?
• 10/22: Begin d>=1. Grids (conforming, affine, shape-regular, acute, quasi-uniform).
• 10/25: Finite element triple (K,P,\Sigma). Example of local stiffness matrix calculations on triangles. {HW4 due 10/22/21}
• 10/27: P_1, Q_1, Crouzeix-Raviart and Raviart-Thomas elements. HW4 discussion part 1.
• 10/29: HW 4 discussion part 2. Practice proofs of coercivity and continuity with alternative norms to H_*. Why [\nabla_n u] is a good error indicator.
• 11/1: Reiterate why conforming V_h requires globally continuous piecewise polynomials, and conforming mesh. Begin studying the opposite situation of DG (Discontinuous Galerkin): 1d version, in several variants.
• 11/3: The need to study Strang I and Strang II lemmas, and the Petrov-Galerkin setup. Quadrature on triangles. BNB1 and BNB2 conditions: examples in finite dimensional spaces.
• 11/5: No class (FE circus)
• 11/8: Finish examples with BNB1 and BNB2. Variational formulation for Petrov-Galerkin case under BNB1 and BNB2, and convergence proof. Revisit and war-up Strang lemmas 1 and 2 {HW3 (group work on software) due 11/8/21}
• 11/10: Projections. Systems of saddle point structure (Darcy, Stokes, incompressible elasticity).
• 11/12: Continue saddle-point problems: from abstract formulation through finite element discretization through linear system.
• 11/15: From primal to dual via Legendre transform. {HW5 due}
• 11/17: The use of inf-sup conditions for mixed Poisson and Stokes problems.
• 11/19: Stability and instability examples for FE discretization of mixed problems.Q1-P0 vs Taylor-Hood.
• 11/22: HW5 discussion. More on projections.
• 11/24: Linear solvers appropriate for FE. From direct through iterative (stationary, non-stationary) to multigrid.
• 11/26: No class (THX holiday)
• 11/29: Proof of L^2 error estimate for a model parabolic problem amd BE discretization in time. Other schemes than BE.
• 12/1: First order terms when using FE: how, and what problems arise. Operator splitting if needed.
• 12/3: Review and group presentations.