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.

翻译：本文提出一种新型高阶不连续伽辽金（DG）方法，该方法无需引入罚参数或稳定化参数，我们将其称为免惩罚DG（\PFDG）。在该方法中，试验函数和检验函数属于间断Sobolev空间，该空间中的函数通常在网格骨架上不连续且不满足Dirichlet边界条件。然而，可在此空间中区分出一个子集，其中函数连续且满足Dirichlet边界条件，该子集称为可允许子集。试验解选自一个*增广*可允许子集，该子集允许连续性条件存在微小违逆。该子集通过对有限元基函数的线性组合施加特殊增广约束而构建。本方法保留了DG方法的所有优势，且无需使用稳定性参数或数值通量。我们针对二维基准问题（泊松方程、线弹性、超弹性及双调和方程）在多边形（三角形、四边形及弱凸多边形）网格上，以及三维六面体网格上的泊松问题进行了数值验证。数值结果表明，该方法在$L^2$范数和能量半范数下具有精确的精度和最优收敛性。