Domain Decomposition Methods - Example 1: 1D Linear BVP

Example 1: 1D Linear BVP

The exact solution is:

Subdivide the domain into two subdomains, one from and another from . In each of these two subdomains define interpolating functions and At the interface between these two subdomains the following inferface conditions shall be imposed:


Let the interpolating functions be defined as:




Where is the nth cardinal function of the chebyshev polynomials of the first kind with input argument y.
If N=4 then the following approximation is obtained by this scheme:







This was obtained with the following MATLAB code.

clear all N=4; a1=0; b1=1/2; =cheb(N,a1,b1); % the diff matrices on are the same %as those on . I=eye(N+1); H=D2-I; H1=; H(2:end-1,:); ]; H1=]]; H2=]; H2= H2]; K=; F=; u=K\F; xx=-cos(pi*(0:N)'/N); x1=1/4*(xx+1); x2=1/4*(xx+3); x=; uex=(exp(x)-exp(-x))./(exp(1)-exp(-1));