Direct discontinuous Galerkin method and its variations for second order elliptic equations (Q520207)

The authors apply the discontinuous Galerkin method to find the approximate solutions of the Dirichlet problem for the second-order elliptic equation \[ - \nabla ( K(x) \nabla u) + c(x)u = f \text{ in }\Omega \subset \mathbb R^{2} \] \[ u = u_{0} \text{ on }\partial \Omega \] using its variations form. The regular partition of the domain \( \Omega \) into disjoint elements \(K\) defines the set \(T_h\). \( P_{l}(K)\) presents the polynomial function space of degree at most \(l\) on the elements \(K\). The direct discontinuous Galerkin method, studied by \textit{H. Liu} and \textit{J. Yan} [SIAM J. Numer. Anal. 47, No. 1, 675--698 (2009; Zbl 1189.65227)], is applied for three different schemes: symmetric, non-symmetric and a scheme with interface correction. Numerical results of this method are given for five examples.