We will present an approach to numerical simulation on recursively structured adaptive discretisation grids. The respective grid generation process is based on recursive bisection of triangles along marked edges. The resulting refinement tree is sequentialised according to a Sierpinski space-filling curve, which leads to both minimal memory requirements and inherently cache-efficient processing schemes. The locality properties induced by the space-filling curve are even retained throughout adaptive refinement of the grid. We demonstrate the efficiency of the approach by implementing a multilevel-preconditioned conjugate gradient solver for a simple, yet adaptive, test problem: solving Poisson’s equation on a re-entrant corner problem.