Optimal error estimates of the local discontinuous Galerkin method for nonlinear second-order elliptic problems on Cartesian grids

In this paper, we study the local discontinuous Galerkin (LDG) methods for two-dimensional nonlinear second-order elliptic problems of the type u_xx + u_yy = f(x, y, u, u_x, u_y), in a rectangular region Ω with classical boundary conditions on the boundary of Ω. Convergence properties for the solution and for the auxiliary variable that approximates its gradient are established. More specifically, we use the duality argument to prove that the errors between the LDG solutions and the exact solutions in the L² norm achieve optimal (p + 1)th order convergence, when tensor product polynomials of degree at most p are used. Moreover, we prove that the gradient of the LDG solution is superclose with order p + 1 toward the gradient of Gauss–Radau projection of the exact solution. The results are valid in two space dimensions on Cartesian meshes using tensor product polynomials of degree p ≥ 1, and for both mixed Dirichlet–Neumann and periodic boundary conditions. Preliminary numerical experiments indicate that our theoretical findings are optimal.