Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

$hp$-optimal convergence of the original DG method for linear hyperbolic problems on special simplicial meshes

Zhaonan Dong, Lorenzo Mascotto

TL;DR

The paper establishes $hp$-optimal convergence for the original discontinuous Galerkin method solving first-order hyperbolic problems with constant convection fields, specifically in the $L^2$ norm and the DG norm, on special simplicial meshes. This is achieved by developing hp-optimal approximation properties of the Cockburn-Dong-Guzmán (CDG) projector across 1D, 2D, and 3D, including sharp trace-type estimates on element boundaries. The authors prove $||u-u_h||_{0,\Omega} \lesssim h^{\min(k,p)+1} p^{-(k+1)} ||u||_{k+1,\Omega}$ and $|||u-u_h|||_{DG} \lesssim h^{\min(k,p)+1/2} p^{-(k+1/2)} ||u||_{k+1,\Omega}$ under admissible-mesh assumptions, and confirm these rates with numerical experiments that remain robust even when some assumptions are mildly violated. The results extend $hp$-convergence theory to the $p$-version on simplicial meshes, complementing prior work on tensor-product grids and informing high-order DG assessments for hyperbolic problems.

Abstract

We prove hp-optimal error estimates for the original DG method when approximating solutions to first-order hyperbolic problems with constant convection fields in the L2 and DG norms. The main theoretical tools used in the analysis are novel hp-optimal approximation properties of the special projector introduced in [Cockburn, Dong, Guzman, SINUM, 2008]. We assess the theoretical findings on some test cases.

$hp$-optimal convergence of the original DG method for linear hyperbolic problems on special simplicial meshes

TL;DR

The paper establishes -optimal convergence for the original discontinuous Galerkin method solving first-order hyperbolic problems with constant convection fields, specifically in the norm and the DG norm, on special simplicial meshes. This is achieved by developing hp-optimal approximation properties of the Cockburn-Dong-Guzmán (CDG) projector across 1D, 2D, and 3D, including sharp trace-type estimates on element boundaries. The authors prove and under admissible-mesh assumptions, and confirm these rates with numerical experiments that remain robust even when some assumptions are mildly violated. The results extend -convergence theory to the -version on simplicial meshes, complementing prior work on tensor-product grids and informing high-order DG assessments for hyperbolic problems.

Abstract

We prove hp-optimal error estimates for the original DG method when approximating solutions to first-order hyperbolic problems with constant convection fields in the L2 and DG norms. The main theoretical tools used in the analysis are novel hp-optimal approximation properties of the special projector introduced in [Cockburn, Dong, Guzman, SINUM, 2008]. We assess the theoretical findings on some test cases.
Paper Structure (14 sections, 6 theorems, 104 equations, 5 figures)

This paper contains 14 sections, 6 theorems, 104 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Admissible meshes and the method
  3. Admissible simplicial meshes
  4. The original DG method
  5. The CDG projector and its approximation properties
  6. Preliminary results
  7. The CDG projector and the main result
  8. The 1D case
  9. The 2D case
  10. The 3D case
  11. Trace-type estimates for the CDG operator
  12. A priori analysis
  13. Numerical experiments
  14. Conclusions

Key Result

Proposition 2.1

Let $\overline c$ be given in standard-assumption-convection. The following bound holds true:

Figures (5)

  • Figure 1: Left-panel: mesh consisting of 50 triangles. Right-panel: mesh consisting of 32 triangles.
  • Figure 2: $p$-version for the case of constant convection field $\boldsymbol\beta$. We consider the exact solution $u$ in \ref{['u1']} and fix the triangular mesh with 50 elements as in Figure \ref{['figure:meshes']} so as the mesh assumptions in Section \ref{['subsection:meshes']} are satisfied. The singularity lies in the interior of a cell element. Left-panel: $L^2$ norm error. Right-panel: DG norm error.
  • Figure 3: $p$-version for the case of constant convection field $\boldsymbol\beta$. We consider the exact solution $u$ in \ref{['u1']} and fix the triangular mesh with 32 elements as in Figure \ref{['figure:meshes']} so as the mesh assumptions in Section \ref{['subsection:meshes']} are satisfied. The singularity lies at the interface of several cell elements. Left-panel: $L^2$ norm error. Right-panel: DG norm error.
  • Figure 4: $p$-version for the case of variable convection field $\boldsymbol\beta$. We consider the exact solution $u$ in \ref{['u1']} and fix the triangular mesh with 50 elements as in Figure \ref{['figure:meshes']}. The mesh assumptions in Section \ref{['subsection:meshes']} are not satisfied. The singularity lies in the interior of a cell element. Left-panel: $L^2$ norm error. Right-panel: DG norm error.
  • Figure 5: $p$-version for the case of variable convection field $\boldsymbol\beta$. We consider the exact solution $u$ in \ref{['u1']} and fix the triangular mesh with 32 elements as in Figure \ref{['figure:meshes']}. The mesh assumptions in Section \ref{['subsection:meshes']} are not satisfied. The singularity lies at the interface of several cell elements. Left-panel: $L^2$ norm error. Right-panel: DG norm error.

Theorems & Definitions (11)

  • Proposition 2.1
  • Remark 1
  • Lemma 3.1
  • Theorem 3.2
  • Proposition 3.3
  • proof
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • ...and 1 more