Introduction

A lot of PDE(s) can be writen in a generic form, and depends mainly on the definition of coefficients. The generic form that we use is describe by the next equation :

\[\begin{eqnarray*} d \frac{\partial u}{\partial t} + \nabla \cdot \left( -c \nabla u - \alpha u + \gamma \right) + \beta \cdot \nabla u + a u = f \quad \text{ in } \Omega \end{eqnarray*}\]

with \(u\) the unknown and \(\Omega \in \mathbb{R}^n\) the computation domain . We call this generic form by Coefficient Form PDE and the coefficients are

\(d\) : damping or mass coefficient
\(c\) : diffusion coefficient
\(\alpha\) : conservative flux convection coefficient
\(\gamma\) : conservative flux source term
\(\beta\) : convection coefficient
\(a\) : absorption or reaction coefficient
\(f\) : source term

Many problems are multiphysics (i.e. several unknowns) and the generic form can be extended naturally into a system of generic PDE.

Coefficients of each equation can be defined by an expression that depends of the current unknown or with unknowns of others equations.

The unknown of each PDEs can be defined as a scalar function (\(u:\mathbb{R}^n \longrightarrow \mathbb{R}\)) or a vectorial function (\(u:\mathbb{R}^n \longrightarrow \mathbb{R}^n\)). But we need to respect some constraint on the coefficient shape as describe in the next table.

Table 1. Shape required by the coefficients
Coefficient	Scalar Unknown	Vectorial Unknown
\(d\)	scalar	scalar
\(c\)	scalar or matrix	scalar or matrix
\(\alpha\)	vectorial	x
\(\gamma\)	vectorial	x
\(\beta\)	vectorial	vectorial
\(a\)	scalar	scalar
\(f\)	scalar	vectorial

Introduction

1. Initial Conditions

2. Boundary Conditions

3. Finite Element Approximation

4. Time scheme

5. Stabilized finite element