| Prima settimana | |
| 4 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. |
| 5 Marzo - Lez | Finite elements for 1D elliptic equation. Piecewise linear finite element. Construction of the linear system. Computation of the integrals by using the reference element. Interpolation error estimates. Higher degree finite elements. |
| 7 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 | |
| 11 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. |
| 12 Marzo - Lez | Generalization to other boundary conditions: non homogeneous Dirichlet boundary conditions and Neumann boundary condition. Advection, diffusion and reaction equation. Weak form and finite element discretization. |
| 14 Marzo - Lab | Solution of 1D elliptic equations with nonhomogeneous Dirichlet boundary conditions and Neumann problem. Solution of an advection dominated equation in 1D. |
| Terza settimana | |
| 18 Marzo - Lez | Parabolic equation. Weak form. Semidiscretization in space and derivation of the system of ordinary differential equations associated with the space discretization. Time advancing scheme: θ-method. |
| 19 Marzo - Lez | Separation of variable for the solution of the heat equation. Eigenvalue problem associated and representation with eigenfunctions. Analysis of the stability of the θ-method. Error estimates for the semidiscrete and the fully discrete solution. |
| 21 Marzo - Lab | Solution of partial differential equations of elliptic type using the PDE toolbox of MATLAB. Examples. |
| 25 Marzo - Lab | Solution of partial differential equations of parabolic type. Behavior of the θ-method for 1D problems. Use of the toolbox of MATLAB for solving problem with two dimensions in space. |