# Finite Difference and Finite Volume as special cases of Finite Element

I have heard this many times that "FD is a special case of FE" and separately, that "FV is a special case of FE".

However I have never come across a brief document that substantiates that claim, through derivation, examples, proofs, whatever.

I understand the basics of FE and the 3 manifestations: Collocation, Least-squares, Galerkin.

I also understand that these three can be considered as various kinds of "weighted residual" schemes, and the unifying principle is MWR (method of weighted residuals) which is a variational method.

Are there two easy-to-read resources, one for FD and one for FV, that can make the connection to either FE or MWR or something like that?

• Actually, that's backwards: MWR is arguably the most general; collocation and Galerkin are special cases; FD is a special case of collocation and FE and FV are special cases of Galerkin. The difference is how you chose the finite-dimensional space for the solution and the finite-dimensional space for the test (or weight) function. (Not everyone would agree on this ordering, depending on your precise definition of Galerkin and MWR. I don't see much value in nailing this down to the level of a formal theorem, either, but it is helpful to realize the common features of the three approaches.) Nov 17, 2017 at 23:27
• I'm not primarily looking for a theorem and proofs. In fact I would prefer if there are as little, or no, theorems and proofs. "Easy to read" means for a scientist/engineer (since I'm not a mathematician, and especially not very good with functional analysis). However, a combination of elaboration, derivations, examples, and putting things in context, is highly desirable. Otherwise those are empty claims/assertions. Nov 17, 2017 at 23:32
• So what specifically are you looking for (assuming it's not just idle curiosity)? The problem is that your premise (first sentence) does not hold up if you look at it in any detail; at most you can say that FE methods are Galerkin methods, some FV methods can be interpreted as generalized (discontinuous) Galerkin methods, and (with some violence), so can some FD methods. Nov 17, 2017 at 23:40
• I saw an explanation of this once upon a time in a big, thick CFD book, but for the life of me, i can’t remember the title... i’ll seeif i can track it down.
– Paul
Nov 18, 2017 at 2:00

A good example is if you use bilinear finite elements for the Laplace equation on a uniform mesh, and then approximate the integrals using the trapezoidal rule, then you get the usual 3-point stencil (in 1d) or 5-point stencil (in 2d) that is well known from finite difference methods. The only difference is that both the left and right hand side of the linear system are multiplied by the square of the mesh size, but other than that, the linear system is exactly the same as the one you'd get from a finite difference method.

• Although once you extend to Poisson's equation, you see an unusual stencil acting on the right hand side. Dec 5, 2017 at 14:46
• Um, why? You still have the right hand side only evaluated at the grid point if you apply the same quadrature formula. Dec 5, 2017 at 22:53
• Apologies, I'm used to exhibiting the result in 1d with exact integration. Dec 5, 2017 at 23:07
• The same happens with linear (triangle in 2D) elements with exact integration of the basis functions $W_{ij} = \int_{\Omega} \nabla \phi_i \cdot \nabla \phi_j$ and a lumped mass matrix - one gets $-\Delta u = f \to W u = Mf$ which is the same as the finite difference system. Jan 13 at 16:42

I can't speak much for FV, never having worked much with the technique. But many FD methods can be rewritten as an FE method that judiciously chooses basis sets and quadrature rules in such a way that the implicit "LHS" matrix is simplified (typically to diagonal form). Some of these techniques have been around a long time under the name "mass lumping".

I wrote about this relationship a bit in my thesis, there's a link below (see especially section 2.4 starting on page 11). It's all couched in terms of solving the Maxwell system using FDTD. In 1D this reduces to just scalar Helmholtz. You can follow the same process for Laplace. For other PDE's, you might have to ask an expert what the appropriate basis sets are.

Title: H-, P- and T-Refinement Strategies for the Finite-Difference-Time-Domain (FDTD) Method Developed via Finite-Element (FE) Principles

• Link in answer expired. Here is the new one etd.ohiolink.edu/acprod/odb_etd/ws/send_file/… Jan 8 at 10:51
• Appreciate the correction, hoisted that up into the body of the answer. Welcome to scicomp! Jan 12 at 15:52
• Thank you @rchilton1980 Jan 15 at 13:21