I try to solve a system of coupled PDEs using FEM. Unfortunately, the originating matrix has very poor condition. After days of double checking and thinking, I suspect the following reason:
Given a PDE
a * Laplacian(u) + b * u = 0
For FEM, one has to calculate two operator matrices, one for Laplacian(u) and one for u.
When L is the element size, as far as I understand, the matrix entries for Laplacian(u) scale with 1 / L while the ones for u scale with L.
My system has a size of several micrometers, but I use meters as unit for lengths. So the elements for Laplacian(u) are very big, while the ones for u are very small.
So, could I improve the matrix condition by using a shorter length unit for the elements, maybe millimeters or micrometers? (I mean not the unit of u, but the unit of the geometry.)