A new discontinuous Galerkin method, conserving the discrete H²-norm, for third-order linear equations in one space dimension

We introduce a Bassi-Rebay type discontinuous Galerkin (DG) method for both stationary and time-dependent third-order linear equations. This method is the first DG method which conserves the mass and the L2-norm of the approximations of the solution and that of its first and second derivatives. For the stationary case, L2-projections of the errors (in the approximation of the solution, its first and second derivatives) are proven to have optimal convergence rates when the polynomial degree k is even and the mesh is uniform, and to converge suboptimally, but sharply, with order k when k is odd or the mesh is nonuniform. We show that suitably defined projections of the errors superconverge with order k+1+min{ k,12 } on uniform meshes and converge optimally on nonuniform meshes. The numerical traces are proven to superconverge with order 2k if k is odd or the mesh is nonuniform. For even k and uniform meshes, we show that the numerical traces superconverge with order 2k+32. If in addition, the number of intervals is odd, the convergence order is improved to 2k+32+min{ k,12 }. This allows us to use an element-by-element postprocessing to construct new approximations that superconverge with the same orders as the numerical traces. For the time-dependent case, the errors are proven to be of order k+1 for even k on uniform meshes and of order k when k is odd or the mesh is nonuniform. Numerical results are displayed, which verify all of the above-mentioned theoretical orders of convergence as well as the conservation properties of the method. We also show that the orders of convergence of the stationary case also hold for the time-dependent case.