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As far as I know, convergence refers to having a solution which results in an error which is less than the target. In a circuit simulator, the routine is trying to solve for the node voltages while trying to make sure that algebraic sum of currents into that node is zero (Kirchoff's Current Law - KCL). Instead of zero, a small number is usually taken, say 10uA. So if the algebraic sum of current is less than or equal to 10uA, the routine may say that it has *converged* to a solution. Of course, I am grossly oversimplifying the routine here. If the system never gets to less than or equal to 10uA, the routine might try to cut down on time step size, etc. If all these fail, you usually get a error message saying there is a *convergence problem*. At this point, you can relax the convergence tolerance to, say, 20uA and try again.
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