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

Abstract

We analyze the discontinuous Galerkin method in time combined with a finite element method with symmetric stabilization in space to approximate evolution problems with a linear, first-order differential operator. A unified analysis is presented for space discretization, including the discontinuous Galerkin method and H 1-conforming finite elements with interior penalty on gradient jumps. Our main results are error estimates in various norms for smooth solutions. Two key ingredients are the post-processing of the fully discrete solution by lifting its jumps in time and a new time-interpolate of the exact solution. We first analyze the L ∞ (L 2) (at discrete time nodes) and L 2 (L 2) errors and derive a super-convergent bound of order (τ k+2 + h r+1/2) for static meshes for k ≥ 1. Here, τ is the time step, k the polynomial order in time, h the size of the space mesh, and r the polynomial order in space. For the case of dynamically changing meshes, we derive a novel bound on the resulting projection error. Finally, we prove new optimal bounds on static meshes for the error in the time-derivative and in the discrete graph norm.

Post-processed Galerkin approximation of improved order for wave equations

“…3) amounts to a result of superconvergence at these time points. The proof of (1.3) strongly differs from the proof developed in [17] for first-order partial differential equations. This is a key point of the analysis of this work.…”

“…The key contribution of this work is the post-processing of the fully discrete space-time finite element solution by lifting it in time from a continuous to a continuously differentiable approximation. For this, a new lifting operator L τ , that is motivated by the work done in [17] for discontinuous Galerkin methods, is introduced. We derive error estimates for the lifted space-time approximation with respect to u, ∇u and ∂ t u in the L 2 (Ω)-norm at all time points t ∈ [0, T ], as well as in the L 2 (0, T ; L 2 (Ω))-norm.…”

“…In [17], a post-processing procedure for a discontinuous Galerkin method in time combined with a stabilized finite element method in space for linear first-order partial differential equations is introduced and analyzed. The post-processing of the fully discrete solution lifts its jumps in time such that a continuous approximation in time is obtained.…”

“…In particular, superconvergence of order τ k+2 + h r+1 2 , measured in the norm of L ∞ (L 2 ) (at the discrete time nodes) and L 2 (L 2 ), is established for static meshes and k ≥ 1. The analysis of [17] strongly depends on a new time-interpolate of the exact solution. The work [17] uses ideas of [33] where a post-processing is developed for variational time discretizations of nonlinear systems of ordinary differential equations.…”

“…The difference in the proofs comes through the stability estimate given in Lemma 5.10. For the second-order hyperbolic problem (1.1), rewritten as a first-order system in time, a weaker stability result compared with [17,Lemma 4.2] is obtained such that in the resulting error analysis some contributions can no longer be absorbed by terms on the left-hand side of the error inequality like in [17]. Therefore, to show (1.3), a completely different approach is developed.…”