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Optimal error estimate of an isoparametric upwind discontinuous Galerkin method for radiation transport equation on curved domains

Changhui Yao, Yunpan Ma, Lingxiao Li

TL;DR

The paper addresses the stationary radiation transport equation $\Omega\cdot \nabla I + \sigma I = f$ in a curved domain $D$ with inflow data on $\Gamma^-$ and develops an isoparametric mapping-based upwind discontinuous Galerkin method to recover the optimal convergence by balancing geometry and discretization errors. It introduces an isoparametric framework with a mapping $F_h$ from a straight reference mesh to the curved domain and an auxiliary mapping to separate geometric error from discretization error, proving an $O(h^{k+1/2})$ convergence in the DG norm for $k\ge 2$. The key contributions include establishing bounds on the isoparametric approximation error $||\tilde{I}-\Lambda\tilde{I}||_{DG}$ and the geometric inflow boundary error, along with a tight consistency error analysis for the upwind DG scheme. Numerical experiments in both 2D and 3D corroborate the theory, showing that curved-domain isoparametric DG achieves the predicted rates and outperforms straight-mesh discretizations, thereby enabling accurate high-order RTE simulations on curved geometries.

Abstract

This work investigates the isoparametric upwind discontinuous Galerkin method for solving the radiation transport equation defined on a bounded domain $D$ with a piecewise $C^{k+1}$ smooth curved boundary. An auxiliary mapping is constructed to approximate the original curved domain. The analysis delineates a high-order optimal convergence rate under the DG norm, which comprehensively balances the errors stemming from the numerical discretization and the geometric approximation. Two- and three-dimensional numerical experiments validate the theoretical results.

Optimal error estimate of an isoparametric upwind discontinuous Galerkin method for radiation transport equation on curved domains

TL;DR

The paper addresses the stationary radiation transport equation in a curved domain with inflow data on and develops an isoparametric mapping-based upwind discontinuous Galerkin method to recover the optimal convergence by balancing geometry and discretization errors. It introduces an isoparametric framework with a mapping from a straight reference mesh to the curved domain and an auxiliary mapping to separate geometric error from discretization error, proving an convergence in the DG norm for . The key contributions include establishing bounds on the isoparametric approximation error and the geometric inflow boundary error, along with a tight consistency error analysis for the upwind DG scheme. Numerical experiments in both 2D and 3D corroborate the theory, showing that curved-domain isoparametric DG achieves the predicted rates and outperforms straight-mesh discretizations, thereby enabling accurate high-order RTE simulations on curved geometries.

Abstract

This work investigates the isoparametric upwind discontinuous Galerkin method for solving the radiation transport equation defined on a bounded domain with a piecewise smooth curved boundary. An auxiliary mapping is constructed to approximate the original curved domain. The analysis delineates a high-order optimal convergence rate under the DG norm, which comprehensively balances the errors stemming from the numerical discretization and the geometric approximation. Two- and three-dimensional numerical experiments validate the theoretical results.
Paper Structure (13 sections, 8 theorems, 60 equations, 2 figures, 4 tables)

This paper contains 13 sections, 8 theorems, 60 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Isoparametric finite element method on curved mesh
  3. Computational domain and auxiliary mapping
  4. Isoparametric finite element spaces of RTE
  5. Error analysis of the isoparametric finite element method
  6. Approximation error in the isoparametric finite element space
  7. The consistency error of the upwind DG method
  8. The difference between $\tilde{I}$ and $I^\dag$
  9. Geometric approximation error of inflow boundary
  10. Numerical Experiment
  11. Example two-dimensional(2D)
  12. Example three-dimensional(3D)
  13. Conclusion

Key Result

Lemma 2.1

Assume that there exist two positive constants $m$ and $M$ such that $0<m\leq\sigma\leq M$. Then the numerical solution $I_h$ of (eq2.4) obeys the bound

Figures (2)

  • Figure 1: Isoparametric mapping and numerical results in 2D. Left: The straight mesh $\mathcal{\widehat{T}}_{h_1}$ (yellow), interpolation nodes (blue), and curved domain boundary (red) after isoparametric mapping with $k=2$. Right: Numerical and theoretical convergence rate on $\mathcal{T}_h$ for $k=2$ and $k=3$.
  • Figure 2: Mesh and domain evolution through isoparametric mapping in 3D. Left: The straight mesh $\widehat{\mathcal{T}}_{h}$ with convex polyhedron domain $\widehat{D}_h$. Middle: The straight mesh $\widehat{\mathcal{T}}_{h}$ (light blue) with curved domain $D_h$ (gray). Right: The curved mesh $\mathcal{T}_{h}$ with curved domain $D_h$.

Theorems & Definitions (10)

  • Lemma 2.1
  • Remark 1
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Remark 2
  • Lemma 3.6
  • Theorem 3.7