2. Generalities on PDE

Most of the physical problems modeled by PDEs fall in one of the following three large classes: equilibrium, propagation, or eigenvalue problems.

• In an equilibrium problem, we are interested in finding a function u∈ Hð Þ Ω such that

$$\mathcal{D}u = f \quad \text{in } \Omega \tag{1}$$

$$
u = 
u\_D \quad \text{on } \Omega\_D \tag{2}$$

$$\frac{\partial u}{\partial \mathbf{n}} = u\_N \quad \text{on } \Omega\_N \tag{3}$$

In (1)–(3) D is an operator that includes derivatives and f ∈ Hð Þ Ω is known. Boundary conditions are typically given as constraint about u or its derivatives on regions of ∂Ω. Typical examples of physical systems giving rise to this type of problem are systems in steady state (e.g., temperature distribution, potential distribution, steady flows, and so on). Typically equilibrium problems are elliptic, that is, D is an elliptic operator (a generalization of the Laplacian).


$$\mathcal{D}u = \lambda u \tag{4}$$

A wide class of eigenvalue problems is represented by Sturm-Liouville problems that can be written as

$$[p \cdot \mathbf{y'}]' + q \cdot \mathbf{y} = -\lambda w \cdot \mathbf{y} \tag{5}$$

where the apostrophe denotes derivation, the unknowns are λ, and function is y∈ Hð Þ ½ � a; b , while p, q, and w, all belonging to Hð Þ ½ � a; b , are known. Sturm-Liouville problems include Bessel differential equations (obtained by writing Laplace, Helmholtz, or Schrodinger equation in cylindrical coordinates) and Lagrange differential equation (obtained working in spherical coordinates).

### 2.1 Solution of differential equations

The field of numerical solution of differential equations is very wide, and many techniques have been developed. Nevertheless, a categorization in few large classes is possible. An important step in every solution algorithm is mapping the differential equation into a discrete version with only a finite number of degrees of freedom. A first distinction can be done between techniques that achieve this objective by discretizing the domain Ω or the function space Hð Þ Ω .
