Many real world problems can be modeled with Partial Differential Equations (PDEs). Since for many PDEs no exact solution can be found, there exists a variety of methods which give an approximate solution to those PDEs. One method which can be applied to find an approximate solution for elliptic PDEs is the Parallel Multilevel Partition of Unity Method (PMPUM). The major computational effort in this method is needed for the discretization of the differential operator. In this work we focus on the applicability of General-purpose computing on graphics processing units (GPGPU) on this task. A GPGPU implementation of the PMPUM is and a comparison to a given CPU implementation is presented. It is shown that the implementation using a GPGPU approach can be applied to many cases arising in the PMPUM to improve the performance.