Discontinuous Galerkin method in time combined with a stabilized finite element method in space for linear first-order PDEs (Q2814433)

The authors consider the discontinuous Galerkin method in time coupled with a finite element method with symmetric stabilization in space for solving evolution problems with a linear, first-order differential operator. They analyze the error between the exact solution and a post-processed discrete solution which is continuous in time and is a piecewise polynomial in time of order of \((k+1)\). By lifting its jumps in time for the post-processing of the fully discrete solution and using a new time-interpolate of the exact solution, error estimates in various norms for smooth solutions are obtained. In particular, they first analyze the \(L^\infty(L^2)\) (at discrete time nodes) and \(L^2(L^2)\) errors and derive a superconvergent bound of order \((\tau^{k+2}+h^{r+1/2})\) for static meshes for \(k\geq 1\). Here, \(\tau\) is the time step, \(k\) the polynomial order in time, \(h\) the size of the space mesh, and \(r\) the polynomial order in space. They further derive a novel bound on the resulting projection error for the case of dynamically changing meshes. New optimal bounds on static meshes for the error in the time-derivative and in the discrete graph norm are finally proved.