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This article analyzes the effect of the penalty parameter used in  symmetric dual-wind discontinuous Galerkin (DWDG) methods for approximating second order elliptic partial differential equations (PDE).  DWDG methods follow from the DG differential calculus framework that defines discrete differential operators used to replace the continuous differential operators when discretizing a PDE. We establish the convergence of the DWDG approximation to a continuous Galerkin approximation as the penalty parameter tends towards infinity. We also test the influence of the regularity of the solution for elliptic second-order PDEs with regards to the relationship between the penalty parameter and the error for the DWDG approximation. Numerical experiments are provided to validate the theoretical results and to investigate the relationship between the penalty parameter and the L^2-error.For more information see https://ejde.math.txstate.edu/conf-proc/26/l1/abstr.html