One of the major weaknesses of the phase field method for fracture is the necessity for using very fine discretization, dictated by the value of length-scale parameter. In literature, it has often been reported that the edge length of finite elements has to be multiple times smaller than the length scale parameters. Too coarse mesh can lead to erroneous model global and local response, such as overestimation of critical forces, or inaccurate force-displacement diagrams and cracking patterns. It is widely believed that this occurs when a finite element mesh is not fine enough to properly capture the phase-field profile. However, in finite element formulations that use a single mesh for both displacements and the phase field the phase field profile exhibits a plateau with a constant value instead of a sharp tip for the fully developed crack profile, which spuriously increases the total fracture energy. In this work, we propose a new discretization approach that relies on utilizing two different finite element meshes for the discretization of displacements and phase field. Here, the phase field mesh consists of polygonal elements constructed such that their nodes are placed at the centers of elements used for the displacement field. Various approaches are presented, and their performance demonstrated by a set of numerical tests. We demonstrate that the proposed method avoids plateaus in the phase field profile and yields improved accuracy and convergence in comparison to the original dual-mesh approach based on finite volumes and classical finite element formulations, especially for sudden crack propagation. For stable crack propagation results suggest that in that case convergence could also be significantly affected by the computation of the crack driving force in highly damaged elements. These findings could open up new directions in designing new efficient adaptive techniques for phase field methods for brittle fracture.