I'm currently writing a report on the Element-free Galerkin method, which is a meshless method to solve PDES. In the method the stiffness matrix K(which is positive definite) has to be modified to impose the boundary conditions. The modification leads to a new matrix, M(defined below) which is not positive definite.
K and G are sparse matrices. Resulting system to solve is
In all the literature I've read this is reported to be a drawback of this particular method, but i don't understand why. We can still solve the final system so whats the difference ?
I'm using Matlab for the method and solving the system using
U = M\F
Thanks
SPD matrices can be solved by the Conjugate Gradient (CG) algorithms and it's variations which are very efficient.
This is because SPD matrices have many very pleasent properties.
Also it is usually quite easy to show that a matrix is SPD, but if it isn't SPD you have to resort to more complicated tools to show that it is invertible, and you have to resort to more computationally expensive solving algorithms. But as long as you can use matlabs backslash operator you don't have to worry about it. It can even recognize non invertible matrices and use pseudo inverses.