We develop two a posteriori error estimates, namely projected a posteriori error estimation and postprocessed a posteriori error estimation, for the continuous Galerkin method for nonlinear initial value problems. The projected a posteriori error estimator exhibits <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10092_2025_638_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {O}(k^r)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mi>k</mi> <mi>r</mi> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> convergence rate in the <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10092_2025_638_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-norm when piecewise r-degree polynomials are used, and is asymptotically exact for the projected <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10092_2025_638_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-error. In addition, the postprocessed a posteriori error estimator is asymptotically exact for the true <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10092_2025_638_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-error. Both error estimators are fully explicit with respect to the local time-steps and approximation orders, and are used to design an hp-adaptive refinement procedure. Several numerical examples are provided to verify the theoretical results and illustrate the efficiency of the hp-adaptive algorithm.

A posteriori error estimates and adaptivity for the continuous Galerkin time-stepping method for nonlinear initial value problems

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发表日期：

2025年4月24日

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Liutao Tian,
Shuo Yang,
Hongjiong Tian

摘要

We develop two a posteriori error estimates, namely projected a posteriori error estimation and postprocessed a posteriori error estimation, for the continuous Galerkin method for nonlinear initial value problems. The projected a posteriori error estimator exhibits $\mathcal {O}(k^r)$ $O (k^{r})$ convergence rate in the $L^2$ $L^{2}$ -norm when piecewise r-degree polynomials are used, and is asymptotically exact for the projected $L^2$ $L^{2}$ -error. In addition, the postprocessed a posteriori error estimator is asymptotically exact for the true $L^2$ $L^{2}$ -error. Both error estimators are fully explicit with respect to the local time-steps and approximation orders, and are used to design an hp-adaptive refinement procedure. Several numerical examples are provided to verify the theoretical results and illustrate the efficiency of the hp-adaptive algorithm.