TY - JOUR
T1 - An $hp$-Version of $C^0$ -Continuous Petrov-Galerkin Time-Stepping Method for Second-Order Volterra Integro-Differential Equations with Weakly Singular Kernels
AU - Li , Shuangshuang
AU - Wang , Lina
AU - Yi , Lijun
JO - East Asian Journal on Applied Mathematics
VL - 1
SP - 20
EP - 42
PY - 2020
DA - 2020/11
SN - 11
DO - http://doi.org/10.4208/eajam.020520.120620
UR - https://global-sci.org/intro/article_detail/eajam/18411.html
KW - $hp$-version, second-order Volterra integro-differential equation, weakly singular kernel,
continuous Petrov-Galerkin method, exponential convergence.
AB - <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>