Direct Discontinuous Galerkin Method and Its Variations for Second Order Elliptic Equations

In this paper, we study direct discontinuous Galerkin method (Liu and Yan in SIAM J Numer Anal 47(1):475-698, 2009) and its variations (Liu and Yan in Commun Comput Phys 8(3):541-564, 2010; Vidden and Yan in J Comput Math 31(6):638-662, 2013; Yan in J Sci Comput 54(2-3):663-683, 2013) for 2nd order elliptic problems. A priori error estimate under energy norm is established for all four methods. Optimal error estimate under norm is obtained for DDG method with interface correction (Liu and Yan in Commun Comput Phys 8(3):541-564, 2010) and symmetric DDG method (Vidden and Yan in J Comput Math 31(6):638-662, 2013). A series of numerical examples are carried out to illustrate the accuracy and capability of the schemes. Numerically we obtain optimal th order convergence for DDG method with interface correction and symmetric DDG method on nonuniform and unstructured triangular meshes. An interface problem with discontinuous diffusion coefficients is investigated and optimal th order accuracy is obtained. Peak solutions with sharp transitions are captured well. Highly oscillatory wave solutions of Helmholz equation are well resolved.