A convection–diffusion model for the averaged flow of a viscous, incompressible magma through an elastic medium is considered. The magma flows through a dike from a magma reservoir to the Earth’s surface; only changes in dike width and velocity over large vertical length scales relative to the characteristic dike width are considered. The model emerges when nonlinear inertia terms in the momentum equation are neglected in a viscous, low-speed approximation of a magma flow model coupled to the elastic response of the rock. Stationary- and traveling-wave solutions are presented in which a Dirichlet condition is used at the magma chamber; and either a (i) free-boundary condition, (ii) Dirichlet condition, or (iii) choked-flow condition is used at the moving free or fixed-top boundary. A choked-flow boundary condition, generally used in the coupled elastic wave and magma flow model, is also used in the convection–diffusion model. The validity of this choked-flow condition is illustrated by comparing stationary flow solutions of the convection–diffusion and coupled elastic wave and magma flow model for parameter values estimated for the Tolbachik volcano region in Kamchatka, Russia. These free- and fixed-boundary solutions are subsequently explored in a conservative, local discontinuous Galerkin finite-element discretization. This method is advantageous for the accurate implementation of the choked flow and free-boundary conditions. It uses a mixed Eulerian–Lagrangian finite element with special infinite curvature basis function near the free boundary and ensures positivity of the mean aperture subject to a time-step restriction. We illustrate the model further by simulating magma flow through host rock of variable density, and magma flow that is quasi-periodic due to the growth and collapse of a lava dome.