@Article{EAJAM-11-1,
author = {Shuangshuang, Li and Wang, Lina and Yi, Lijun},
title = {An $hp$-Version of $C^0$ -Continuous Petrov-Galerkin Time-Stepping Method for Second-Order Volterra Integro-Differential Equations with Weakly Singular Kernels},
journal = {East Asian Journal on Applied Mathematics},
year = {2021},
volume = {11},
number = {1},
pages = {20--42},
abstract = {<p style="text-align: justify;">An $hp$-version of $C^0$-CPG time-stepping method for second-order Volterra
integro-differential equations with weakly singular kernels is studied. In contrast to
the methods reducing second-order problems to first-order systems, here the CG and
DG methodologies are combined to directly discretise the second-order derivative. An
a priori error estimate in the $H^1$-norm, fully explicit with respect to the local discretisation and regularity parameters, is derived. It is shown that for analytic solutions with
start-up singularities, exponential rates of convergence can be achieved by using geometrically refined time steps and linearly increasing approximation orders. Theoretical
results are illustrated by numerical examples.</p>},
issn = {2079-7370},
doi = {https://doi.org/10.4208/eajam.020520.120620},
url = {https://global-sci.com/article/82512/an-hp-version-of-c0-continuous-petrov-galerkin-time-stepping-method-for-second-order-volterra-integro-differential-equations-with-weakly-singular-kernels}
}