We present performance results from a new hybridized finite difference method for the spatial discretization of partial differential equations. The method is based on the standard Summation-By-Parts method with weak enforcement of boundary and interface conditions through the Simultaneous-Approximation-Term. We analyze the performance when applying the hybrid method to Poisson's equation which arises in many steady-state physical problems, focusing on an application in Earth science. When solving the resulting linear system we compare direct and iterative solvers on both CPU and GPU, evaluating the performance on meshes with different numbers of computational blocks. Our results demonstrate the advantages of using the hybrid method in solving large-scale problems under the restriction of system resources by utilizing techniques from parallel computing.