# Quantify difference between regularity and irregularity

I am solving an equation numerically on a 1D-domain using the finite-element method. I am solving it using two different domains, one regular and one irregular. Naturally, the solution varies slightly between the two and I would like to quantify how much.

So, how would be a good way to quantify the difference in the solutions vs. the difference in the two domains? Keywords and hints are greatly appreciated.

• A good choice might depend on the physics. An appropriate mathematical choice would probably be some integral norm. In 1D FEM such integral norms are straightforward to compute. Specifically, you just represent the difference in your finite element basis, where you use elements which are the refinement of the two meshes into one another. For instance, if one was $0,1,2,3,4,5$ and the other was $0,0.7,1.6,3.2,4.1,5$, then the refinement would be $0,0.7,1,1.6,2,3,3.2,4.1,5$. (This is harder to do in higher dimensions.) – Ian Aug 25 '15 at 10:51
• @Ian Thanks. So this is a graphical way to characterize it, where I simply merge the data sets and plot them. Is it possible to get a number that would characterize the "smoothness"/regularity of the solution given a "smoothness"/regularity of the mesh? – BillyJean Aug 25 '15 at 11:04
• While you can do it graphically, that's not actually what I meant. You have two finite element solutions $f,g$ on different grids. You merge the grids, and represent $f-g$ on the new grid. (This works because if, for example, $f$ and $g$ are piecewise linear on their original grids, then $f-g$ is also piecewise linear on the new grid.) Then you compute some integral norm of $f-g$, e.g. $\int_D |f(x)-g(x)| dx$. As for regularity, that's already getting quite a bit more problem-specific. You might try using a Sobolev norm instead of a Lebesgue norm for that. – Ian Aug 25 '15 at 11:20
• @Ian Thanks, you definitely gave me a good push. If you have time/ideas, I am very interested in how I can further quantify the regularity, hints/keywords etc.. I'll look into Sobolev spaces. – BillyJean Aug 25 '15 at 12:14