<p>In this paper, we establish energy-stable and maximum principle preserving in a discrete sense for the exponential time differencing (ETD) schemes of the Allen–Cahn equation with the skeletal finite element method (FEM), which is hybridized discontinuous Galerkin (HDG) FEM with the Lehrenfeld-Schöberl stabilization. The schemes employ the first- and second-order ETD method in the time direction and the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_3057_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$P_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>-<InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_3057_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$P_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> skeletal FEM in the space direction, respectively. We demonstrate that the first- and second-order ETD schemes of the skeletal FEM preserve the discrete maximum principle and discrete energy stable unconditionally on triangular meshes. The optimal convergence order estimates in both <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_3057_Article_IEq3.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> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_3057_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$H^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation> norms are obtained. Some numerical results are presented to validate the theory of discrete energy stability, discrete maximum principle and error estimates.</p>

Energy Stability and Maximum Principle of Skeletal Finite Element Method for Allen–Cahn Equation with Exponential Time Differencing Schemes

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发表日期：

2025年9月13日

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Mingze Qin,
Qilong Zhai,
Ran Zhang

摘要

In this paper, we establish energy-stable and maximum principle preserving in a discrete sense for the exponential time differencing (ETD) schemes of the Allen–Cahn equation with the skeletal finite element method (FEM), which is hybridized discontinuous Galerkin (HDG) FEM with the Lehrenfeld-Schöberl stabilization. The schemes employ the first- and second-order ETD method in the time direction and the $P_1$ $P_{1}$ - $P_0$ $P_{0}$ skeletal FEM in the space direction, respectively. We demonstrate that the first- and second-order ETD schemes of the skeletal FEM preserve the discrete maximum principle and discrete energy stable unconditionally on triangular meshes. The optimal convergence order estimates in both $L^2$ $L^{2}$ and $H^1$ $H^{1}$ norms are obtained. Some numerical results are presented to validate the theory of discrete energy stability, discrete maximum principle and error estimates.