An ℎ𝑝-version Legendre-Jacobi spectral collocation method for Volterra integro-differential equations with smooth and weakly singular kernels

Wang, Zhong-qing; Guo, Yu-ling; Yi, Li-jun

doi:10.1090/mcom/3183

An $hp$-version Legendre-Jacobi spectral collocation method for Volterra integro-differential equations with smooth and weakly singular kernels
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by Zhong-qing Wang, Yu-ling Guo and Li-jun Yi PDF

Math. Comp. 86 (2017), 2285-2324 Request permission

Abstract:

In this paper, we present an $hp$-version Legendre-Jacobi spectral collocation method for Volterra integro-differential equations with smooth and weakly singular kernels. We establish several new approximation results of the Legendre/Jacobi polynomial interpolations for both smooth and singular functions. As applications of these approximation results, we derive $hp$-version error bounds of the Legendre-Jacobi collocation method under the $H^1$-norm for the Volterra integro-differential equations with smooth solutions on arbitrary meshes and singular solutions on quasi-uniform meshes. We also show the exponential rates of convergence for singular solutions by using geometric time partitions and linearly increasing polynomial degrees. Numerical experiments are included to illustrate the theoretical results.

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