Superconvergence of UWLDG Method for One-Dimensional Linear Sixth-Order Equations

This paper concerns the superconvergence property of the ultraweak-local discontinuous Galerkin (UWLDG) method for one-dimensional linear sixth-order equations. The crucial technique is the construction of a special projection. We will discuss in three different situations according to the remainder of k, the highest degree of polynomials in the function space, divided by 3. We can prove the \((2k-1)\) $(2k-1)$ th-order superconvergence for the cell averages when \(k\equiv 0\) $k\equiv 0$ or 2 (mod 3). But if \(k\equiv \) $k\equiv$ 1 (mod 3), we can only prove a \((2k-2)\) $(2k-2)$ th-order superconvergence. The same superconvergence orders can also be gained for the errors of numerical fluxes. We will also prove the superconvergence of order \(k+2\) $k+2$ at some special quadrature points. Some numerical examples are given at the end of this paper.