High‐order continuous and discontinuous Galerkin methods for wave problems

Abstract

hree Galerkin methods—continuous Galerkin, Compact Discontinuous Galerkin, and hybridizable discontinuous Galerkin—are compared in terms of performance and computational efficiency in 2-D scattering problems for low and high-order polynomial approximations. The total number of DOFs and the total runtime are used for this correlation as well as the corresponding precision. The comparison is carried out through various numerical examples. The superior performance of high-order elements is shown. At the same time, similar capabilities are shown for continuous Galerkin and hybridizable discontinuous Galerkin, when high-order elements are adopted, both of them clearly outperforming compact discontinuous GalerkinPeer ReviewedPostprint (published version

Hybridizable Discontinuous Galerkin with degree adaptivity for the incompressible Navier–Stokes equations

“…Its influence is crucial and has been discussed and analyzed for a large number of problems by Cockburn and co-workers see, for instance, [20][21][22][23][24][25]. Choosing the correct stabilization parameter should provide sufficient stabilization to the solution and, at the same time, should ensure superconvergence of the post-processed solution, see also the discussion in [12].…”

“…While in the diffusion limit (no convection) the single face τ guarantees both stability and superconvergence with simple numerical considerations (see [12]), in presence of convection this choice is not available. In fact, the single face approach is not capable to stabilize the convective effect, see [23] for a detailed discussion.…”

“…Note that the post-process solution is not required to be computed at each time step, but only when an improved solution is needed. Moreover, the computational overhead is small and decreases with the approximation degree, see [12].…”

“…This normalization is crucial for non-uniform meshes. Note that the evaluation of the improved velocity u * requires an inexpensive element-by-element computation, see [12] and Table 1 in examples Section 4.2. Moreover, the post-process, which implies evaluating u * , is only done when the error is evaluated, not necessarily at each time step.…”

“…On the other hand, the ability and efficiency of high-order approximations has been discussed and proven in recent papers [11][12][13]. In particular, the interest in Discontinuous Galerkin (DG) methods has increased over the past years, because they have proved their suitability to construct robust stabilized high-order numerical schemes on arbitrary unstructured and non-conforming grids for a variety of physical phenomena, see, for instance, [14][15][16][17][18][19][20] for the solution of incompressible flows.…”