In general, the material presented in the lectures should be regarded as pensum, except:
- The lectures 'Error estimates and adaptivity' (ie, lectures 15 and 16)
- The proof of convergence for parabolic problems
- The lecture on continuum mechanics (number 12)
- Multidimensional convection-diffusion problems and later (ie lectures 22 and 23)
| Week | Topic | Reading | Additional reading |
|---|---|---|---|
| 34 | Introduction to finite element methods and weak solutions. 1D Poisson equation. | Quarteroni 3.2, 4.1-3, Lecture 1 Lecture 2 | H1, H2, H3 |
| 35 | PDE theory: weak solutions, Sobolev spaces. Poisson equation in higher dimension. | Quarteroni 3.1-4. Sobolev spaces | Quarteroni 2 |
| 36 | 2D Poisson equation 1: Weak and Galerkin formulations. Lagrange basis, barycentric coordinates and reference elements. Assembly by quadrature and transformation to reference element | Quarteroni 3.3, 4.1, 4.4-5, 8.2 Lecture 4 Lecture 5 | H4 |
| 37 | 2D Poisson equation 2: Assembly (continued). Implementation of boundary conditions. Stability and convergence of approximation | Quarteroni 3.3-4, 4.1-2, 4.5, 8.4 Lecture 6 Lecture 7 | H4 |
| 38 | General elliptic and parabolic problems | Quarteroni 3.4, 5.1 | |
| 39 | Parabolic equations: Weak and Galerkin formulations with theta-method time discretization. Stability and convergence of approximations in space and time | Quarteroni 5.1-5 Lecture 9 Lecture 10 | |
| 40 | Conditioning of stiffness matrix, equivalence of L2 norms. Continuum mechanics: stress, strain and linear elasticity equations | Quarteroni 4.5.2, Lecture 11 Lecture 12 | Stress Strain Linear Elasticity |
| 41 | Grids: Delaunay triangulation and advancing front method. Spacing functions and 1D adaptivity | Quarteroni 6 Lecture 13 Lecture 14 | |
| 42 | Error estimates and adaptivity: L2 and L-infinity error estimates. A priori adaptivity with derivative reconstruction. A posteriori adaptivity and spacing functions | Quarteroni 4.6, 6, Lecture 15 Lecture 16 | |
| 43 | Numerical linear algebra: Classical iterative methods, conjugate gradient and Krylov subspace methods | Quarteroni 7, Lecture 17 Lecture 18 | |
| 44 | Numerical linear algebra 2: Multigrid methods. Convection-diffusion problems: first comments and 1D illustrative cases | Quarteroni 7, 12.1-4 Lecture 19 Lecture 20 | |
| 45 | Convection-diffusion problems 2: Lumping of mass matrix. Artificial diffusion and generalized Galerkin framework. Galerkin least squares and other regularizations. | Quarteroni 12.5-8 Lecture 21 Lecture 22 | |
| 46 | Conclusion: Brief outlook | Lecture 23 | |
| 47 | Revision |