This paper focuses on high-accuracy numerical methods for second-order weakly singular Volterra integro-differential equations. We propose and analyze a novel hp-version <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$C^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-Continuous Petrov-Galerkin (CPG) scheme for such equations. This method employs piecewise <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$C^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-continuous polynomials as the trial space and piecewise discontinuous polynomials of two degrees lower as the test space, which allows the scheme to be implemented as a time-stepping method. We rigorously establish the well-posedness of the proposed scheme and derive optimal estimates in the <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$H^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$H^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> norms. An essential aspect of these estimates is that the generic constant C does not depend on both the time step sizes and polynomial degrees. Furthermore, for solutions exhibiting an initial singularity, we demonstrate that the CPG method achieves exponential convergence when employing geometrically graded meshes in combination with linearly increasing polynomial degrees. Numerical results are also provided to verify the theoretical estimates.

hp-version $C^1$-CPG method with exponential convergence for weakly singular Volterra integro-differential equations

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发表日期：

2026年5月2日

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Fujun Liu,
Lina Wang,
Lijun Yi

摘要

This paper focuses on high-accuracy numerical methods for second-order weakly singular Volterra integro-differential equations. We propose and analyze a novel hp-version $C^1$ $C^{1}$ -Continuous Petrov-Galerkin (CPG) scheme for such equations. This method employs piecewise $C^1$ $C^{1}$ -continuous polynomials as the trial space and piecewise discontinuous polynomials of two degrees lower as the test space, which allows the scheme to be implemented as a time-stepping method. We rigorously establish the well-posedness of the proposed scheme and derive optimal estimates in the $L^2$ $L^{2}$ , $H^1$ $H^{1}$ , and $H^2$ $H^{2}$ norms. An essential aspect of these estimates is that the generic constant C does not depend on both the time step sizes and polynomial degrees. Furthermore, for solutions exhibiting an initial singularity, we demonstrate that the CPG method achieves exponential convergence when employing geometrically graded meshes in combination with linearly increasing polynomial degrees. Numerical results are also provided to verify the theoretical estimates.

hp-version \(C^1\)-CPG method with exponential convergence for weakly singular Volterra integro-differential equations

摘要