Abstract In this paper we discuss numerical methods and software tools for solving partial differential equations with variable local resolution. The emphasis of our work is on finding approaches which will greatly simplify developing applications using dynamic adaptive grids. First, we investigate a class of adaptive grids that looks particularly promising for use in general-purpose solvers and propose a discretization method suitable for this class of grids. Second, we describe an interactive programming environment which can be used, among other things, as a geometrically-oriented debugger for developing applications using adaptive grids. Finally, we briefly illustrate an application of the above-mentioned methods and software to flame–flow interaction problems.