Optimal convergence of local discontinuous Galerkin methods for convection-diffusion equations

Abstract

The $hp$ local discontinuous Galerkin (LDG) method proposed by Castillo et al. [Math. Comp.,~71 (238): 455-478, 2002] has been shown to be an efficient approach for solving convection-diffusion equations. However, theoretical analysis indicates that, for solutions with limited spatial regularity, the method exhibits suboptimal convergence in $p$, suffering a loss of one order, comparing to numerical experiments. The purpose of this paper is to close the gap between theoretical estimates and numerical evidence. This is accomplished by establishing new approximation results for the associated Gauss-Radau projections of functions in suitable function spaces that can optimally characterize the regularity of singular solutions. We show that such a framework arises aturally and enables the study of various types of singular solutions, with full consistency between theoretical analysis and numerical results. This investigation sheds light on the resolution of the suboptimality in $p$ observed in the literature for several other types of DG schemes in different settings.

Figures (3)

• Figure 2.1: Error plots of the $p$-version LDG for the singular solution $u(x,t)=x^\pi t$ with the convection coefficient: $c=0.1$ and the diffusion coefficients: $d=0.1$ and $d=0$. Left: Snapshot of Castillo2002MC. Right: Recovery of the same results with slopes of the error plots.
• Figure 4.1: The convergence of the $p$-version for the non-smooth exact solutions, where $u(x,t)=x^{\pi} e^{2+\sin(x)-t}$ (left), and $u(x,t)={\mathcal{I}}_{0+}^{\pi} H(\zeta-x) \,t$ with $\zeta = (x_0 + x_1)/2$ (right). The convection coefficient is $c=0.1$ and the diffusion coefficient is $d=0.1$ and $d=0.$
• Figure 4.2: The convergence of the $p$-version for the nonsmooth exact solutions $u(x,t)=|x-\zeta|^{\pi} e^{2+\sin(x)-t}$, where $\zeta=x_1$ on the left side and $\zeta= (x_0 + x_1)/2$ on the right side. The convection coefficient is $c=0.1$ and the diffusion coefficient is $d=0.1$ and $d=0.$

Theorems & Definitions (4)

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