We investigate discrete counterparts of antidynamo theorems for advection–diffusion of differential forms, a framework that can be regarded as a generalized Hodge theory. Within finite element exterior calculus (FEEC), we propose a finite element discretization of the Lie advection–diffusion operator and analyze the structure of its stationary solutions. We demonstrate that the space of stationary discrete k-forms is closely connected to the topology of the underlying manifold. Specifically, under suitable assumptions, the dimension of this space is governed by the k-th Betti number for sufficiently large diffusivity. Furthermore, we establish sharper, diffusivity-independent conclusions for the specific cases of k=0, 1, n-1, and n. We also examine conditions under which practical numerical schemes satisfy the assumptions of the analysis. The results provide discrete counterparts of classical anti-dynamo theorems and clarify how topological properties are retained at the discrete level.