FEM Weak formulation verification

[wish17]

Hi everyone,

I am writing my own FEM program solving a 2-D diffusion equation for work. My initial version of the code is working okay; however, the stiffness matrix is very ill-conditioned which I suspect is resulting in inaccurate solution. I suspect I have issues in the weak formulation of the equation as well as in the application of the boundary conditions. Please see the attached derivation.

The weak form of the equation is given in Eq(3). When I evaluate the RHS of (3) by using the boundary conditions at z=1 and z=0, I get that it goes to zero. This gives me the following matrix equation to solve: [K]*{G} = {f}, where {f} is a zero vector, resulting in null solution vector {G}.

Any feedback on my derivation or the proper method of the application of boundary conditions would be much appreciated.

Thanks!

 

[rb1957]

did you try googling "solving illconditioned matrix" ?

another day in paradise, or is paradise one day closer ?

 

[wish17]

Yes, I did! One of the common reasons for the stiffness matrix to be ill-conditioned is due to improper application of the boundary conditions. I think this might be the case for me as well but I can't seem to figure out why my approach is incorrect. That is why I am asking for any feedback at all here.

 

[rb1957]

in your equation 2 … you've copied too much of eqn 1 (no "= 0" on the LHS, inside the brkts).

also in eqn 3 … why limits of integration for z are 0 to zH (and not 0 to 1) ?

why do you think your boundary conditions are "improper" ? what do your boundary conditions mean (in words) ?
you're modelling a disc? (r = a)



another day in paradise, or is paradise one day closer ?