Additive Schwarz Method - Overview

Overview

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.

(Model problem) The heat distribution in a square metal plate such that the left edge is kept at 1 degree, and the other edges are kept at 0 degree, after letting it sit for a long period of time satisfies the following boundary value problem:

f_xx(x,y) + f_yy(x,y) = 0

f(0,y) = 1; f(x,0) = f(x,1) = f(1,y) = 0

where f is the unknown function, f_xx and f_yy denote the second partial derivatives with respect to x and y, respectively.

Here, the domain is the square × .

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.