  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.
