TY - JOUR
T1 - On Discontinuous and Continuous Approximations to Second-Kind Volterra Integral Equations
AU - Liang , Hui
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 1
SP - 91
EP - 124
PY - 2022
DA - 2022/02
SN - 15
DO - http://doi.org/10.4208/nmtma.OA-2021-0141
UR - https://global-sci.org/intro/article_detail/nmtma/20223.html
KW - Volterra integral equations, collocation methods, Galerkin methods, discontinuous
Galerkin methods, convergence analysis.
AB - <p style="text-align: justify;">Collocation and Galerkin methods in the discontinuous and globally continuous piecewise polynomial spaces, in short, denoted as DC, CC, DG and CG
methods respectively, are employed to solve second-kind Volterra integral equations
(VIEs). It is proved that the quadrature DG and CG (QDG and QCG) methods obtained from the DG and CG methods by approximating the inner products by suitable
numerical quadrature formulas, are equivalent to the DC and CC methods, respectively. In addition, the fully discretised DG and CG (FDG and FCG) methods are
equivalent to the corresponding fully discretised DC and CC (FDC and FCC) methods. The convergence theories are established for DG and CG methods, and their
semi-discretised (QDG and QCG) and fully discretized (FDG and FCG) versions. In
particular, it is proved that the CG method for second-kind VIEs possesses a similar
convergence to the DG method for first-kind VIEs. Numerical examples illustrate the
theoretical results.</p>