We present a fully coupled solver based on the discontinuous Galerkin method for steady‐state diffusion flames using the low‐Mach approximation of the governing equations with a one‐step kinetic model. The nonlinear equation system is solved with a Newton–Dogleg method and initial estimates for flame calculations are obtained from a flame‐sheet model. Details on the spatial discretization and the nonlinear solver are presented. The method is tested with reactive and nonreactive benchmark cases. Convergence studies are presented, and we show that the expected convergence rates are obtained. The solver for the low‐Mach equations is used for calculating a differentially heated cavity configuration, which is validated against benchmark solutions. Additionally, a two‐dimensional counter diffusion flame is calculated, and the results are compared with the self‐similar one dimensional solution of said configuration.