Discontinuous Galerkin (DG) methods are extensions of the usual Galerkin finite element methods. Although there are vast amount of studies on DG methods, most of them have assumed shape-regularity conditions on meshes for both theoretical error analysis and practical computations. In this paper, we present a new symmetric interior penalty DG scheme with a modified penalty term. We show that, without imposing the shape-regularity condition on the meshes, the new DG scheme inherits all of the good properties of standard DG methods, and is thus robust on anisotropic meshes. Numerical experiments confirm the theoretical error estimates obtained.

间断Galerkin（DG）方法是常规Galerkin有限元方法的扩展。尽管对DG方法的研究很多，但大多数方法都在网格上假设形状规则性条件，以进行理论误差分析和实际计算。在本文中，我们提出了一种新的带有修正惩罚项的对称内部惩罚DG方案。我们表明，在不对网格施加形状规则性条件的情况下，新的DG方案继承了标准DG方法的所有良好特性，因此在各向异性网格上具有鲁棒性。数值实验证实了所获得的理论误差估计。