Choosing a basis for finite element method

Question

While choosing a basis for a finite element approximation, what is considered a "good" basis? For instance, how does Legendre Polynomials compare to basis of sine functions?

Thanks in advance.

Answer 1

TL;DR: It depends on what you want to solve, but I'd say polynomials are more versatile.

Intro

First I'll say some high-level comments on what you want out of a basis, and then I'll say some words about what makes a good polynomial basis for a single element. This is a very large topic, with a lot of research done on it, so I will inevitably miss some important points. Broadly speaking you want:

1. the basis to span a space that closely approximates the solution
2. a basis element to "behave reasonably"
3. efficient algorithms for common computations.

I'll say some more about the first two of these, and then go into the question you specifically asked about trig functions vs. polynomials.

Choosing a good space

In order for the finite element method to work, you want the exact solution to be close (in some metric) to your approximation space. For example, if you are trying to simulate fluids, you want to be able to capture boundary layers and regions with shocks. This can be done by refining the mesh in regions of interest. A more serious example is the importance of choosing the right space for simulating incompressible materials / fluids. If you choose piecewise linear elements, then your mesh can lock and your solution space won't even be close to the exact solution, no matter how much you refine. In these cases (of saddle-point problems), you want to satisfy the inf-sup conditions, and I'll say no more about that here.

Just like refining the mesh sometimes doesn't help, sometimes improving the polynomial order doesn't help. If you know that your solution will only be once differentiable, it doesn't help to use high-order elements. Instead, you want to use linear or at most quadratic elements, and a finer mesh. So in this case in particular, piecewise-polynomial elements are far better than trigonometric functions. (One way to see this is that the solution isn't very smooth, so the Fourier coefficients won't decay very rapidly, so you need high frequencies to capture the effects.) The other nasty thing that happens when you try to use high-order elements to approximate irregular solutions is the Gibbs phenomena, which can cause numerical instability. (http://en.wikipedia.org/wiki/Gibbs_phenomenon)

So for (1), you want a space that has the right polynomial order of approximation, and mesh refinement in the right areas. This is a relatively uncontroversial statement, but there is a lot of freedom here (some advocate really high order polynomials, some use adaptive mesh refinement).

Choosing a good basis

Here I'll assume you decided to go with a piecewise-polynomial space, which some say is the defining characteristic of FEM. But there's still the choice of making a good basis. Choosing a good basis can improve the numerical stability of your algorithm, and the condition number of your mass / stiffness matrices. As a trivial example of what can go wrong, you could choose each basis polynomial to be nonzero on every element, and this would make your matrices dense.

Now I'll further assume that you are going with a nodal basis, that is we're using Legendre polynomials for some set of points in the element. The key issue here is optimizing the Lebesgue constant (http://en.wikipedia.org/wiki/Lebesgue_constant_%28interpolation%29). Roughly speaking, it says how much changing the value at one node oscillates the solution throughout the element. In particular, the Lebesgue constant controls how well nodal interpolation approximates a function.

If you're doing a low order (up to about cubic) approximation, these issues don't matter very much, so I wouldn't worry about it.

Trig functions vs. Polynomials

I'll be brief here. If you're solving a linear PDE on a torus with a smooth initial condition, then trig functions are a no-brainer. If you're doing nonlinear elasticity on a complicated domain, then I'd advocate using a piecewise quadratic solution. In general, I'm a fan of using p-refinement (that is, using higher order polynomials) as long as it's feasible --this has many of the approximation properties of trig functions, but in some cases simpler algorithms-- and then refining the mesh wherever the solution looks more interesting.

Choosing a good basis is a very large part of FEM, so it's good to play around and see what happens. For more info I might suggest Tom Hughes' book, which I find very accessible. MIT's OCW also has some good resources, almost every department has some sort of FEM class. I think 16.920 is a good class for learning more.