This is an example code I wrote for the deal.ii finite element library's code gallery. It is a large-scale distributed solver for Poisson equation in 2D and 3D solver that uses the local discontinuous Galerkin method. It mainly uses the deal.ii C++ library and the Trilinos library for distributed linear algebra using MPI. I thought it would be helpful for others to have access to this example that goes through writing a discontinuous Galerkin method from scratch and also shows how to do it in a distributed setting using the Trilinos library, both of which were not talked about in other examples in the deal.ii library. This code may also be of interest to users that wish to use the LDG method, as the method is distinctly different from the Interior Penalty Discontinuous Galerkin (IPDG) methods and was not covered in other tutorials on DG methods.