In mathematics, the additive Schwarz method, named after Hermann Schwarz, solves a boundary value problem for a partial differential equation approximately by splitting it into boundary value problems on smaller domains and adding the results.
Partial differential equations (PDEs) are used in all sciences to model phenomena. For the purpose of exposition, we give an example physical problem and the accompanying boundary value problem (BVP). Even if the reader is unfamiliar with the notation, the purpose is merely to show what a BVP looks like when written down.
Here, the domain is the square [0,1] × [0,1].
This particular problem can be solved exactly on paper, so there is no need for a computer. However, this is an exceptional case, and most BVPs cannot be solved exactly. The only possibility is to use a computer to find an approximate solution.
A typical way of doing this is to sample f at regular intervals in the square [0,1] × [0,1]. For instance, we could take 8 samples in the x direction at x = 0.1, 0.2, ..., 0.8 and 0.9, and 8 samples in the y direction at similar coordinates. We would then have 64 samples of the square, at places like (0.2,0.8) and (0.6,0.6). The goal of the computer program would be to calculate the value of f at those 64 points, which seems easier than finding an abstract function of the square.
There are some difficulties, for instance it is not possible to calculate fxx(0.5,0.5) knowing f at only 64 points in the square. To overcome this, one uses some sort of numerical approximation of the derivatives, see for instance the finite element method or finite differences. We ignore these difficulties and concentrate on another aspect of the problem.
Whichever method we choose to solve this problem, we will need to solve a large linear system of equations. The reader may recall linear systems of equations from high school, they look like this: