I have solved a $2D$ PDE for $u(x,z,t)$ using both COMSOL and Matlab. I want to check the convergence of these two methods and try to compare them to each other.
To do this, I ran my Matlab script for various discretization grids: $$ N=64,M=32 \qquad N=128,M=32 \qquad N=256,M=32 $$ $$ N=64,M=64 \qquad N=128,M=128 \qquad N=256,M=256 $$ and saved the results at each $0.1$ seconds of time leaving me with $3D$ arrays in Matlab which I called $mov1, ..., mov6$. For example, in Matlab syntax, $mov1(:,:,1)$ is the $2D$ solution at time $t=0$ and $mov1(M/4+1,:,1)$ is the $1D$ solution at $z=0.25,t=0$.
My thought was to use these to examine the norm of the solution at the cutline $z=0.25$ for both the Comsol solution and for all Matlab solutions $mov1,...,mov6$. What I did was simply take the cutline solution I desired and then used the Matlab $norm$ function. However, what I noticed is that in all cases they seemed to be straight lines with very similar slopes but different $y-intercepts$.
I can't quite explain why but I was able to get them all to match up if I simply standardized a size (say $N=256$; note the size of $M$ doesn't matter because I'm essentially taking this out by only looking at the solution on the specific cutline) and then multiplied all other norm values by $\sqrt(256)/\sqrt(X)$ where $X$ is the size of the $x$-direction in the other solution.
I am wondering if this is a reasonable thing to do. If so, why is this the case? If not, what would be the alternative thing to do?
For reference the graphs I created are below:
Also, as a final comment, the COMSOL solution uses a triangular mesh with a somewhat arbitrary number of discretization points $1609$ in the $x$-direction. So there isn't a simple way to just get the points to line up without using an interpolation method which I would like to avoid doing for the moment.