Maximum principle for discretizations of the heat equation

Question

The maximum principle for the heat equation states that if $u$ is a solution of the heat equation $u_t = \Delta u$, then $u$ attains its maximum on the boundary of its domain.

Suppose $u^n$ is the solution of a discrete version of the heat equation, e.g. a graph heat equation (heat equation using the Laplacian matrix of a graph) or a numerical scheme for the heat equation. When does the maximum principle hold? And how do we prove it?

I'm looking at the proof of the maximum property in the Evan's PDE book (Theorem 4 in sec 2.3.3). It uses the mean value theorem, which doesn't hold on discrete domains.

Answer 1

You need the the discrete counterpart of the maximum principle, aka the discrete maximum principle. See e.g.,

• Ciarlet. Discrete maximum principle for finite-difference operators. Aequationes mathematicae, 1970.

One way to prove this is to show the discretized Laplacian is an $M$-matrix, e.g., see

• Xu and Zikatanov. "A monotone finite element scheme for convection-diffusion equations." Mathematics of Computation 1999.