I'm trying to solve the steady state of a heat equation problem in 2D $\Delta u = 0$ (3D also), with the method of solving the huge system of equations that arises from the discretization of the domain. My boundary conditions are different from any example that I've been able to find.
If my domain is a 2D square, I want to put a fixed temperature (say $T_e$) all around the square edges. However, I want to force a point in the center of the square to be at a different temp from the edges, $T_c, T_c > T_e$. Think of a square metal plate being heated at the center by a soldering iron. I want to calculate the steady state heat in such a scenario.
To the best of my knowledge it seems that this problem is not really a heat equation problem: having a point at the center of the domain with a local maxima temperature goes against the properties of the heat equation...
I'm obtaining results, but it seems that heat does not propagate linearly from the center to the edges, but rapidly decaying. The problem is the same if moved into the 3D domain (a sphere/cube that is heated from it's center to its exterior edges, and my results are the same.
Maybe I'm using a wrong model? or a wrong tool to solve this steady state problem? Any hints or directions will be highly appreciated.