The trade off between accuracy and runtime in finite element (FEM) computations is controlled by the resolution of the underlying mesh. Often, a simulation is first run on a coarse initial mesh which is then refined in regions where the error estimation is above a prescribed tolerance. The most common mesh structures are based on triangle or quad elements which have two issues when refined adaptively: (1) In order to avoid degeneration of elements, uniform subdivision operators are usually applied. This, however, limits the degree of refinement to square numbers or powers of 4 which may cause over-refinement where the tolerance violation before refinement is not strong. (2) Furthermore, the local refinement leads to so-called „hanging nodes“ on the edges along which refined and unrefined elements are adjacent. Fixing hanging nodes requires the refinement of neighboring cells even if their error is below the threshold. This again creates over-refinement and makes it necessary to recompute elements meant to stay unchanged. In this work we explore adaptive refinement of polygonal meshes with cells that can be arbitrary convex polygons. We use power diagrams to represent those meshes. Refinement and adaptation can then be achieved by simply adding new sites to a power diagram or by adjusting the weights of the sites. Our approach adapts cell sizes based on an error function obtained from an initial FEM simulation. This allows us to freeze those regions in the mesh that do not need any further refinement after the initial computation, while other regions use the local error estimates to predict the required refinement level and hence the number of sites that need to be inserted in the initial mesh. This way we avoid the iterative refinement where cells are further split if the error reduction after the first split was not sufficient.