We perform a comparison of mass conservation properties of the continuous (CG) and discontinuous (DG) Galerkin methods on non-conforming, dynamically adaptive meshes for two atmospheric test cases. The two methods are implemented in a unified way which allows for a direct comparison of the non-conforming edge treatment. We outline the implementation details of the non-conforming direct stiffness summation algorithm for the CG method and show that the mass conservation error is similar to the DG method. Both methods conserve to machine precision, regardless of the presence of the non-conforming edges. For lower order polynomials the CG method requires additional stabilization to run for very long simulation times. We addressed this issue by using filters and/or additional artificial viscosity. The mathematical proof of mass conservation for CG with non-conforming meshes is presented in Appendix B.
Kopera, M. A., & Giraldo, F. X. (2015). Mass conservation of the unified continuous and discontinuous element-based Galerkin methods on dynamically adaptive grids with application to atmospheric simulations. Journal of Computational Physics, 297, 90–103. https://doi.org/10.1016/j.jcp.2015.05.010