# Non-overlaping Domain decomposition - assemble of Laplacian

I am dealing with following 2-dimensional problem in the unit square domain $S_2$

$$- \Delta u (x,y) = f \ \text{in} \ S_2, \hspace{1.5cm} u(x,y) = 0 \ \text{on} \ \partial S_2$$

where $f$ is such the analytical solution to problem is

$$u(x,y) = \cos [ 2 \pi x (y - x) ]^2.$$

Problem should be discretized by 5-point finite difference scheme, which will result in Ax = b system. I need to decompose problem into 4 non-overlaping domains. Should I divide my domain square into 4 parts and then assemble? or assemble and then divide? Main question here is, how the laplacians will look like.

• Why do you have to decompose the problem into 4 non-overlapping domains? Is that part of the problem description somehow? – Lukas Bystricky Aug 5 '15 at 14:49
• @HH No it is my choice. It can be as many domains as I want, but they have to be non-overlaping. I have chosen 4, since it is a 2D square it is easy to do cut in middle of every dimension. – bla_bla_bla Aug 5 '15 at 16:45