Significance of Sobolev spaces for numerical analysis & PDEs?

Question

I never had an option to take a Functional Analysis module. I am tied up with other work for the next two months so I won't get a chance to self-study it until September. So one thing I was wondering about is the significance of Sobolev spaces for the fields of numerical analysis and PDEs. I have been told on more than one occasion that they are very important in these fields.

Having not taken Functional Analysis, I've never encountered Sobolev spaces before. Would someone be able to give me an overview of what is so significant about these spaces and why are they are so relevant to the above fields?

Answer 1

Sobolev spaces are useful because they are complete function spaces with a norm that

1. reflects the differentiability of functions (unlike $L^p$ norm)
2. has nice geometry (unlike $C^k$ norm)
3. allows approximation by $C^\infty$ functions (unlike $C^k$ norm)

"Nice geometry" means: uniformly convex norm (often, even inner-product norm). This property gives reflexivity which in turn yields

1. Concrete representation of linear functionals. This enables reformulation of problems using duality.
2. Weak compactness of closed bounded sets. With compactness arguments one can show the existence of extremals in variational problems.

Even problems that are not obviously variational at first can be usefully treated as such (like solving $Ax=b$ sometimes turns into minimization of $\|Ax-b\|^2$).

Approximation by $C^\infty$ functions makes it possible to prove estimates for smooth functions first, using the machinery of derivatives, and then extend to the whole space by density.

Answer 2

Suppose you want to find a number $r$ whose square $r^{2}$ is $2$. That has no meaning for numerical analysis because all numbers on a computer are rational, and $\sqrt{2}$ is not rational. It wasn't until the late 1800's that Mathematicians found a logically consistent way to define a real number. But once such a beast could be defined, then one can prove that various algorithms will get you closer and closer to $r$ to $\sqrt{2}$, knowing that it has something to converge to. The existence of such a thing in the extended "real" number system became important to the discussion.

Sobolev spaces are to the ordinary differentiable functions what the real numbers are to the rational numbers. In the late 1800's it was discovered that Calculus of variations didn't have minimizing or maximizing functions. It was the same type of problem: a larger class of functions had to be considered, and the corresponding definitions of integrals had to be extended in order to make sense of and to find a unique minimizer or maximizer that would solve variational problems. So new functions spaces emerged, Lebesgue integration extended the integral expressions to new function classes, and solutions could be found. Once minimizing or maximizing functions could be found, their properties could be deduced, and it validated various algorithms used to find solutions that couldn't converge before because there was nothing to converge to.