<p>This article proposes a new formulation of the adaptive finite element and finite difference methods to obtain an approximate solution to the Riesz-Caputo space-time fractional partial differential equations. We propose the targeted algorithm for complexity addressing in one dimensional nonuniform meshes. The proposed technique uses a known gradient recovery method with optimal accuracy: the polynomial preserving recovery technique, and offers adaptivity. This procedure is based on extensive analytical results about error margins, stability criteria, etc. To emphasize its efficiency even more, the article gives numerous numerical examples showing the algorithm has advantages over the other numerical approaches. This shows the method’s efficiency and a useful implementation for these kinds of fractional partial differential equations posed in fractional calculus.</p>