  In this paper, we present a new high-order discontinuous Galerkin (DG)
method, in which neither a penalty parameter nor a stabilization parameter is
needed. We refer to this method as penalty-free DG (\PFDG). In this method, the
trial and test functions belong to the broken Sobolev space, in which the
functions are in general discontinuous on the mesh skeleton and do not meet the
Dirichlet boundary conditions. However, a subset can be distinguished in this
space, where the functions are continuous and satisfy the Dirichlet boundary
conditions, and this subset is called admissible. The trial solution is chosen
to lie in an \emph{augmented} admissible subset, in which a small violation of
the continuity condition is permitted. This subset is constructed by applying
special augmented constraints to the linear combination of finite element basis
functions. In this approach, all the advantages of the DG method are retained
without the necessity of using stability parameters or numerical fluxes.
Several benchmark problems in two dimensions (Poisson equation, linear
elasticity, hyperelasticity, and biharmonic equation) on polygonal (triangles,
quadrilateral and weakly convex polygons) meshes as well as a three-dimensional
Poisson problem on hexahedral meshes are considered. Numerical results are
presented that affirm the sound accuracy and optimal convergence of the method
in the $L^2$ norm and the energy seminorm.
