<p>The stability and error analysis of a non-uniform implicit-explicit Alikhanov finite element method (IMEX-Alikhanov-FEM) are investigated for a class of time-fractional linear partial differential/integro-differential equations. These equations involve a non-self-adjoint elliptic operator with variable coefficients in both space and time. A second-order error estimate in the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-norm is derived, up to a logarithmic factor, for the problem with initial data in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$H_0^1(\Omega )\cap H^2(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>H</mi> <mn>0</mn> <mn>1</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> <mo>∩</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Furthermore, in the case of a self-adjoint elliptic operator, a superconvergence result is obtained in the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$H^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-norm, leading to an error estimate in the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L^{\infty }$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation>-norm for two-dimensional problems. All the estimates are <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-robust, i.e., constants do not blow up as <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\alpha \rightarrow 1^{-}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo stretchy="false">→</mo> <msup> <mn>1</mn> <mo>-</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>. A set of numerical experiments is conducted to confirm our theoretical results.</p>

A second order non-uniform IMEX-Alikhanov-FEM for time-fractional PDEs and PIDEs with time-dependent coefficients

期刊论文
发表日期：

2025年2月20日

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Aditi Tomar,
Lok Pati Tripathi,
Amiya Kumar Pani

摘要

The stability and error analysis of a non-uniform implicit-explicit Alikhanov finite element method (IMEX-Alikhanov-FEM) are investigated for a class of time-fractional linear partial differential/integro-differential equations. These equations involve a non-self-adjoint elliptic operator with variable coefficients in both space and time. A second-order error estimate in the $L^2$ $L^{2}$ -norm is derived, up to a logarithmic factor, for the problem with initial data in $H_0^1(\Omega )\cap H^2(\Omega )$ $H_{0}^{1} (Ω) \cap H^{2} (Ω)$ . Furthermore, in the case of a self-adjoint elliptic operator, a superconvergence result is obtained in the $H^1$ $H^{1}$ -norm, leading to an error estimate in the $L^{\infty }$ $L^{\infty}$ -norm for two-dimensional problems. All the estimates are $\alpha $ $α$ -robust, i.e., constants do not blow up as $\alpha \rightarrow 1^{-}$ $α \to 1^{-}$ . A set of numerical experiments is conducted to confirm our theoretical results.