FEM: Obtaining the Weak Form

Question

In the the Finite Element Method (FEM), we attempt to obtain the Weak Form of the described equation. I understand that this is an attempt to reduce the order regularity of the equation, but what are the actual benefits and disadvantages of using the Weak Form? What does regularity mean in this sense?

I've checked out other questions on this topic, the math presented there is higher than my level. If I'd been a better mathematician, I suppose I might have derived my answer. Basically, an answer understood by engineers would be nice.

Answer 1

The weak form of the PDE means that you have, in some sense, "fewer constraints on the kinds of functions that you can use" to construct a solution.

To put it into perspective, recall that in FEM, you look for a solution of the form:

$$\sum_{j}u_j*\phi_j(x)$$

where $u_j$ are the unknown scalar coefficients (which will be solved for) and $\phi_j$ are some known functions. The main question is: what kind of functions $\phi_j$ are we allowed to choose?

Well... after going through a very long, tedious, and mathematically pedantic proof, we arrive at a very nice conclusion which is very convenient for engineers who want to implement FEM:

1. All $\phi_j(x)$ must have a value of zero on the boundary.

2. If the weak form of the PDE has a weak derivative of maximum order $k$, then it is sufficient that the functions $\phi_j(x)$ have continuity of order $k-1$.

Condition #1 is very easy to understand: $\phi_j(x)=0$ on all points along the boundary of the domain of your problem.

Condition #2 is not entirely obvious (also not 100% mathematically or pedantically correct either; just "close enough for engineers' sakes"). But condition #2 is the most useful part because it reveals the continuity constraints on $\phi_j$. For example, take the poisson equation in weak form (assuming homogeneous dirichlet boundary conditions)

$$\int_\Omega k\nabla u \cdot \nabla v dx=\int_\Omega fv .$$

This equation has a weak derivative of maximum order k=1 because the gradient here is, effectively, a first order weak derivative (if the weak form had a laplacian operator $\nabla^2$, then k=2; etc...). Thus, you can choose a function space such that $\phi_j(x)$ has continuity of order zero. In other words, you can use functions that are continuous, but not necessarily smooth, across the entire domain. This gives you a lot of flexibility in being able to construct a set of functions statisfying condtion 1.

Why does this matter at all? Because, in FEM, we also seek a third desireable (albeit not absolutely necessary) condition:

3. We want the stiffness matrix to be as sparse as possible.

This third condition matters because it implies that we should use functions $\phi_j$ that are not only zero on the boundary, but zero "almost" everywhere except in small subregions of the domain. Each subregion corresponding to a given $\phi_j$ should only overlap with a small number of other subregions corresponding to other functions.

We can use this condition while exploiting with condition #2 to produce a desirable choice of functions $\phi_j$. Lagrange polynomials are one such set of functions which satisfies all three conditions. Of course, Lagrange polynomials are not the only choice, but are probably the simplest to understand from a beginners' perspective. These functions are zero everywhere except in a small region where they are piecewise linear, quadratic, cubic, etc... Because $\phi_j(x)$ are only non-zero in small regions for each given $j$, the integrals forming the stiffness matrix are also mostly zero and thus the stiffness matrix will also be very sparse.

I just want to caution you that I've watered down a lot of mathematics here and that Condition #2 is only "approximately correct enough for an engineer's perspective" and is not 100% mathematically correct. I leave it to the abundance of mathematically inclined users on this site to point out how gross an overstatement condition #2 is in reality.

Answer 2

It's precisely the opposite of what you had. You use the weak form to increase the regularity (or alternatively, allow you to solve problems with less regularity). Integration always smooths.

The intuition is actually quite simple: For "most" functions, if you take an integral, then its derivative exists (its derivative is the thing you just integrated!). Many times you might want to solve a PDE which doesn't necessarily have derivatives, so the idea is that you just solve the PDE for the integral.

A common example deals with step functions. A step function is not differentiable, but in a sense its "weak derivative" is the Dirac delta (as in, the integral of a step function has a derivatives which integrate like a Dirac delta), so to make everything work you just take integrals everywhere.

Now you can have a PDE for things which aren't strictly differentiable, and you just tell people "well if you look at the integral it works". That's the weak form.

[Note: There are a lot more details. Example, Sobolev Spaces are a generalization of Lebesgue spaces. Since $L^2$ functions aren't functions but equivalence classes of functions which are equal almost everywhere according to some measure, they don't have point-wise values, and so they only make sense in terms of how they integrate (again by how integrals "smoothen"). Etc. But it's all the say idea: smoothen using the integral, and define the derivative via integration by parts.]