<p>We consider the a posteriori error estimation for convection-diffusion-reaction equations in both diffusion-dominated and convection/reaction-dominated regimes. For convection-dominated problems, the energy norm plus a dual norm of convective derivative introduced by Verfürth in 2005 is used to measure the error. We present an explicit hybrid estimator, which, in each regime, is proved to be reliable and efficient with constants independent of the parameters in the underlying problem. For the convection/reaction-dominated regime, we derive a posteriori error estimates that are <i>robust</i> simultaneously to two parameters <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="211_2025_1462_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">$\epsilon ,\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϵ</mi> <mo>,</mo> <mi>β</mi> </mrow> </math></EquationSource> </InlineEquation> that indicate the strength of diffusion and reaction, respectively, under fixed convection. This is proved by showing that the hybrid estimator admits equivalent estimates to the residual estimator introduced by Verfürth in 2005 with robust constants independent of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="211_2025_1462_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">$\epsilon , \beta $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϵ</mi> <mo>,</mo> <mi>β</mi> </mrow> </math></EquationSource> </InlineEquation> and mesh-size. The result generalizes the robust equivalence between hybrid and residual estimators from elliptic interface problems to convection-diffusion–reaction problems. Numerically, various experiments are performed to first demonstrate the robustness of the hybrid estimator and second show that the hybrid estimator is not only more accurate, but also less sensitive to the variation of parameters compared to the residual estimator.</p>

Hybrid a posteriori error estimators for conforming finite element approximations to stationary convection-diffusion-reaction equations

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发表日期：

2025年3月14日

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Difeng Cai,
Zhiqiang Cai

摘要

We consider the a posteriori error estimation for convection-diffusion-reaction equations in both diffusion-dominated and convection/reaction-dominated regimes. For convection-dominated problems, the energy norm plus a dual norm of convective derivative introduced by Verfürth in 2005 is used to measure the error. We present an explicit hybrid estimator, which, in each regime, is proved to be reliable and efficient with constants independent of the parameters in the underlying problem. For the convection/reaction-dominated regime, we derive a posteriori error estimates that are robust simultaneously to two parameters $\epsilon ,\beta $ $ϵ, β$ that indicate the strength of diffusion and reaction, respectively, under fixed convection. This is proved by showing that the hybrid estimator admits equivalent estimates to the residual estimator introduced by Verfürth in 2005 with robust constants independent of $\epsilon , \beta $ $ϵ, β$ and mesh-size. The result generalizes the robust equivalence between hybrid and residual estimators from elliptic interface problems to convection-diffusion–reaction problems. Numerically, various experiments are performed to first demonstrate the robustness of the hybrid estimator and second show that the hybrid estimator is not only more accurate, but also less sensitive to the variation of parameters compared to the residual estimator.