Fluid modelling of an edge plasma is usually performed using a finite-difference scheme on a fixed structured grid. However, both experimental measurement and numerical simulation show the presence of front-like regions characterized by sharp variations of the main plasma parameters such as temperature and radiation power. This is caused in part (i) by strong nonlinearities in the fluid equation coefficients due to abrupt changes of various plasma reaction rates as a function of temperature and (ii) by high anisotropy of the plasma transport along and across magnetic field lines. Manual mesh adoption is usually applied to allow better resolution of the regions with sharp gradients. However, such an approach is very time-consuming and limited. To overcome this problem, we propose to use adaptive unstructured meshes constructed with a new quasi-one-dimensional adaption algorithm. This approach is fast and conservative because we use a new finite-volume scheme. The price of adaptation is high, because numerical algorithms became much more complicated. To avoid unwanted complexity, we suggest an alternative use of a grid-free method, which requires no connectivity of arbitrarily placed vertices. To benchmark the methods and codes in two dimensions, we find analytical and semi-analytical solutions of the nonlinear diffusion–radiation equation, which may have sharp fronts, unconnected boundaries and bifurcated solutions. We use these solutions to study the efficiency of the proposed numerical algorithms.