Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A posteriori error analysis of a space-time hybridizable discontinuous Galerkin method for the advection-diffusion problem

Yuan Wang, Sander Rhebergen

TL;DR

The paper develops a residual-based a posteriori error estimator for a space-time HDG discretization of the time-dependent advection-diffusion problem. It establishes reliability and local efficiency by leveraging a $Pe$-robust coercivity framework and a saturation assumption to control time-derivative errors, enabling fully local space-time adaptivity for problems with boundary and interior layers. The authors provide rigorous proofs and validate them with numerical experiments on rotating Gaussian pulses and layer-type problems, showing that AMR guided by the estimator achieves optimal convergence in the asymptotic regime while exhibiting expected nonrobustness pre-asymptotically. The practical impact lies in enabling robust, layer-resolving adaptive simulations for time-dependent advection-diffusion processes using space-time HDG methods.

Abstract

We present and analyze an a posteriori error estimator for a space-time hybridizable discontinuous Galerkin discretization of the time-dependent advection-diffusion problem. The residual-based error estimator is proven to be reliable and locally efficient. In the reliability analysis we combine a Peclet-robust coercivity type result and a saturation assumption, while local efficiency analysis is based on using bubble functions. The analysis considers both local space and time adaptivity and is verified by numerical simulations on problems which include boundary and interior layers.

A posteriori error analysis of a space-time hybridizable discontinuous Galerkin method for the advection-diffusion problem

TL;DR

The paper develops a residual-based a posteriori error estimator for a space-time HDG discretization of the time-dependent advection-diffusion problem. It establishes reliability and local efficiency by leveraging a -robust coercivity framework and a saturation assumption to control time-derivative errors, enabling fully local space-time adaptivity for problems with boundary and interior layers. The authors provide rigorous proofs and validate them with numerical experiments on rotating Gaussian pulses and layer-type problems, showing that AMR guided by the estimator achieves optimal convergence in the asymptotic regime while exhibiting expected nonrobustness pre-asymptotically. The practical impact lies in enabling robust, layer-resolving adaptive simulations for time-dependent advection-diffusion processes using space-time HDG methods.

Abstract

We present and analyze an a posteriori error estimator for a space-time hybridizable discontinuous Galerkin discretization of the time-dependent advection-diffusion problem. The residual-based error estimator is proven to be reliable and locally efficient. In the reliability analysis we combine a Peclet-robust coercivity type result and a saturation assumption, while local efficiency analysis is based on using bubble functions. The analysis considers both local space and time adaptivity and is verified by numerical simulations on problems which include boundary and interior layers.
Paper Structure (21 sections, 14 theorems, 195 equations, 8 figures)

This paper contains 21 sections, 14 theorems, 195 equations, 8 figures.

Table of Contents

  1. Introduction
  2. The advection-diffusion problem
  3. The space-time hybridizable discontinuous Galerkin method
  4. The space-time mesh
  5. The discretization
  6. A posteriori error analysis
  7. Useful inequalities, approximations and projections
  8. Saturation assumption and time derivative error estimation
  9. Reliability of the error estimator
  10. Local efficiency of the error estimator
  11. Numerical Examples
  12. A rotating Gaussian pulse test
  13. A boundary layer test
  14. An interior layer test
  15. Conclusions
  16. ...and 6 more sections

Key Result

Theorem 4.1

Let $u$ solve eq:advdif, $\boldsymbol{u} = (u, u|_{\Gamma})$, and let $\boldsymbol{u}_h$ solve eq:st_hdg_adr_compact. Assuming that $\delta t_{\mathcal{K}} = \mathcal{O}(h_K^2)$, we have the following reliability estimate

Figures (8)

  • Figure 1: Depiction of sets $\omega_{\mathcal{K}}$, $\sigma_{\mathcal{K}}$, and $\omega_F$ on conforming and 1-irregularly refined meshs. Figures (A), (B), (C): elements in the set $\omega_{\mathcal{K}}$ are the grey colored elements excluding the hatched elements; elements in the set $\sigma_{\mathcal{K}}$ are colored grey and include the hatched elements. Figures (D), (E), (F): elements in the set $\omega_F$ are colored grey.
  • Figure 2: In the left-hand side of the figure we show a $(1+1)$-dimensional example of constructing the subgrid while the right-hand side of the figure gives a $(2+1)$-dimensional illustration of the new facets and edges resulting from the subgrid refinement.
  • Figure 3: Illustration of subgrid projection $i_h^{\mathcal{F}}$ onto an interior $\mathcal{Q}$-facet in $\mathcal{F}_{\mathcal{Q},{h}}$. The neighboring elements of the $\mathcal{Q}$-facet are on the same refinement level in the left column and are on different refinement levels in the right column.
  • Figure 4: The spatial mesh and the rotating pulse. The solution is shown for $\varepsilon=10^{-4}$. Plots correspond to time levels $t = 0.2,0.5,0.8$ from left to right.
  • Figure 5: Convergence histories of the rotating pulse test case. From left to right: $\varepsilon=10^{-3}$, $\varepsilon=10^{-4}$ and $\varepsilon=10^{-5}$. Top row: $\delta t_{\mathcal{K}}=h_K$; middle row: $\delta t_{\mathcal{K}}=\mathcal{O}(h_K^2)$; bottom row: efficiency index for both $\delta t_{\mathcal{K}}=h_K$ and $\delta t_{\mathcal{K}}=\mathcal{O}(h_K^2)$.
  • ...and 3 more figures

Theorems & Definitions (31)

  • Remark 1
  • Theorem 4.1: Reliability
  • Theorem 4.2: Efficiency
  • Remark 2
  • Lemma 4.3
  • proof
  • Lemma 4.4: Local quasi-interpolation estimates
  • proof
  • Lemma 4.5
  • proof
  • ...and 21 more