Abstract We develop and analyze an hp-version of the discontinuous Galerkin time-stepping method for linear Volterra integral equations with weakly singular kernels. We derive a priori error bound in the L 2 -norm that is fully explicit in the local time steps, the local approximation orders, and the local regularity of the exact solutions. For solutions with singular behaviour near t = 0 caused by the weakly singular kernels, we prove optimal algebraic convergence rates for the h-version of the discontinuous Galerkin approximations on graded meshes. Moreover, we show that exponential rates of convergence can be achieved for solutions with start-up singularities by using geometrically refined time steps and linearly increasing approximation orders. Numerical experiments are presented to illustrate the theoretical results.

中文翻译：

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具有弱奇异核的 Volterra 积分方程的不连续 Galerkin 时间步长方法的 hp 版本

摘要 我们开发并分析了具有弱奇异核的线性 Volterra 积分方程的不连续 Galerkin 时间步长方法的 hp 版本。我们在 L 2 -范数中推导出先验误差界限，该界限在局部时间步长、局部逼近阶数和精确解的局部规律中是完全明确的。对于由弱奇异核引起的在 t = 0 附近具有奇异行为的解，我们证明了分级网格上不连续伽辽金近似的 h 版本的最佳代数收敛率。此外，我们表明，通过使用几何细化的时间步长和线性增加的近似阶数，可以为具有启动奇点的解实现指数收敛速度。给出了数值实验来说明理论结果。