Optimal error estimates of the ultra weak local discontinuous Galerkin method for nonlinear sixth-order boundary value problems

In this paper, we focus on studying the convergence and superconvergence properties of ultra weak local discontinuous Galerkin method for nonlinear sixth-order ordinary differential equation of the form −u6(x)+fx,u=0 in one dimension. Firstly, we rewrite the nonlinear sixth-order equation as a third-order system. Subsequently, through integration by parts, all spatial derivatives are transferred to the test functions in the weak formulation. The optimal error estimates for the solution and its third derivative are derived in the L2-norm on arbitrary regular meshes. When using piecewise polynomials of degree up to k, we design special projections to prove the optimal error estimates of order k+1 for the primary solution and the auxiliary variable approximating the third derivative of the solution in the L2-norm. The order of the superconvergence of numerical solutions toward the projections of the exact solutions is proved to be 2k−1, when using the Pk polynomials with k≥3. Numerical experiments are presented to validate the optimal order of accuracy for the proposed error estimates and superconvergence analysis.