FEM, CFD, CAE, and all that.

## Tag: Sobolev space

### What’s the difference?

Today I was asked for a one-sentence comment of the difference between finite difference method and finite element method. What I came up was:

In finite difference method we approximate the differential expression, or the partial derivative, while in finite element method we approximate the space in which the solution lies.

Hopefully this shed some light on the issue.

### Sobolev space: I

Sobolev space $W_p^l$ is based on generalize derivatives and the corresponding norm induced. As a special case, space $H^l$ has the meaning of finite “intensity” in general physical sense. For example, if $u(x)$ is the flow velocity in spatial domain $\Omega$, then $u\in H^1(\Omega)$ means the function and its first order derivatives are all in $L^2(\Omega)$, i.e.

$\int_\Omega{\nabla u \cdot \nabla u+u^2}d\Omega$

Physically it means the total intensity of the sources within $\Omega$ is finite, so is the total kinetic energy within the domain. Obviously this requirement is fundamental for any engineering analysis.

Meyers and Serrin proved that $W_p^l\bigcap C^\infty$ is dense in $W_p^l$ under the  Sobolev space norm.  This means that the function in Sobolev space can be approximated using smooth functions, and in many occasions the properties of Sobolev space is obtained by first demonstrate it in $C^\infty$.

The fundamental question of approximating the variational form of a PDE is twofold:

How much the smoothness/differentiability one can have from the weak solution?

How close the boundary condition can be approximated?

The first question is referred to as regularity problem. The properties of embedding, and traces of the Sobolev spaces are fundamental to answer questions above. It must be emphasized that the regularity of a weak solution $Lu=f$ depends on the data $f$, the geometry of domain, and the boundary condition. All three factors must be examined in practical finite element convergence analysis.