I was curious about researching different methods of solving Laplace's equation and was wondering if there is anything on generating curves on the domain where the solution is constant (contours). For example, in solving first order nonlinear pdes (conservation laws), one can sketch a solution geometrically by finding the characteristic curves of the solution and the value of the solution along these curves. I was wondering if there might be a way of doing something like this for a harmonic function in some bounded and connected domain.

Thanks for any feedback.