| Prima settimana | |
| 5 Marzo - Lez | Second order elliptic partial differential equation. Weak formulation. Existence, uniqueness and stability of the solution. Approximation by finite dimensional subspace. Galerkin method. Céa's Lemma. |
| 6 Marzo - Lez | Finite elements for 1D elliptic equation. Piecewise linear finite element. Interpolation error estimates. Higher degree finite elements. |
| 8 Marzo - Lab | Coding finite elements in 1D. Construction of the stiffness matrix and of the right hand side vector. Use of the reference element. Quadrature formula. Computation of the solution for simple academic examples. Rate of convergence. |
| Seconda settimana | |
| 12 Marzo - Lez | Finite elements for 2D elliptic problems with homogeneous boundary conditions. Interpolation error. Regularity of the solution. Error estimates. Condition number of the stiffness matrix. |
| 13 Marzo - Lez | Generalization to other boundary conditions: non homogeneous Dirichlet boundary conditions and Neumann boundary condition. Weak form and energy estimates for parabolic equations. Semidiscretization with finite elements. |
| 15 Marzo - Lab | Solution of 2D elliptic equations with PDE Toolbox of Matlab. |
| Terza settimana | |
| 19 Marzo - Lez | Time advancing scheme: θ-method. Stability. Error estimates for the semidiscrete and the fully discrete solution. |
| 20 Marzo - Lez | A posteriori error estimators. Adaptive finite element schemes. |
| 22 Marzo - Lab | Error estimates for the finite element discretization of elliptic equations in 2D. Adaptive scheme for singularly perturbed problems or singular solutions. |