Partial differential equation (Rev #2)

The weak formulation of a given PDE is used to relax regularity requirements of the involved functions in order to get weak solutions even when strong solutions do not exist. To get a weak formulation from a strong, conventional formulation of a PDE is a standard, four step procedure. To illustrate it, we choose a model problem:

Let $\Omega \subset \mathbb{R}$ be an open, bounded with a Lipschitz-continuous boundary, our model problem consists of the equation

$- \nabla (a \nabla u) + b u = f$

with given functions $a, b, f$, with $a \ge Const \gt 0$ and $b \gt 0$.

We impose homogeneous Dirichlet boundary conditions:

$u(x) = 0 \; \text{on} \; \partial \Omega$

We derive the weak formulation of the model problem in four steps:

1) Multiply the equation with a test function $v \in C_0^{\infty}(\Omega)$:

$- (\nabla (a \nabla u)) v + b u v = f v$

2) Integrate over $\Omega$:

$- \int_{\Omega} (\nabla (a \nabla u)) v \; d x + \int_{\Omega} b u v \; d x = \int_{\Omega} f v \; d x$

3) Use a appropriate version of the general Stoke’s theorem and the boundary conditions to simplify the equation, that is to reduce the order of the differentials. In our case we can use Green’s theorem and the vanishing of the test function $v$ on the boundary to get

$\int_{\Omega} a \nabla u \; \nabla v \; d x + \int_{\Omega} b u v \; d x = \int_{\Omega} f v \; d x$

4) Find for every involved function the biggest function space such that all integrals converge. This will often result in a function space for candidate solutions $u$ that are bigger and require less regularity than for the original formulation of the PDE.

- Partial differential equation, Wikipedia

- Weak formulation, Wikipedia

category: mathematical methods