AN ARBITRARY-ORDER ULTRA-WEAK DISCONTINUOUS GALERKIN METHOD FOR TWO-DIMENSIONAL SEMILINEAR SECOND-ORDER ELLIPTIC PROBLEMS ON CARTESIAN GRIDS

In this paper, we present and analyze a new ultra-weak discontinuous Galerkin (UWDG) finite element method for two-dimensional semilinear second-order elliptic problems on Cartesian grids. Unlike the traditional local discontinuous Galerkin (LDG) method, the proposed UWDG method can be applied without introducing any auxiliary variables or rewriting the original equation into a system of equations. The UWDG scheme is presented in details, including the definition of the numerical fluxes, which are necessary to obtain optimal error estimates. The proposed scheme can be made arbitrarily high-order accurate in two-dimensional space. The error estimates of the presented scheme are analyzed. The order of convergence is proved to be p + 1 in the L²-norm, when tensor product polynomials of degree at most p and grid size h are used. Several numerical examples are provided to confirm the theoretical results.