East Asian J. Appl. Math., 11 (2021), pp. 20-42.
Published online: 2020-11
Cited by
- BibTex
- RIS
- TXT
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.
}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.020520.120620}, url = {http://global-sci.org/intro/article_detail/eajam/18411.html} }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.