I want to solve a matrix system of the form $Ax=b$ where $A$ is ill-conditioned. The matrix system comes from a structural simulation problem which was discretized using finite elements. I do not have access to the finite element model. I know that proper boundary conditions improve the condition number of the matrix but this should not be the point here.
The matrix system should be solved using an iterative method like CG. What can I do numerically to improve the convergence speed and the accuracy of the solution or even get a useful solution?