Date:
Speaker : Zhongqiang Zhang, Department of Mathematical Sciences, Worcester Polytechnic Institute
Time : 15:00 - 16:00 CET (Rome/Paris)
Hosted at: SISSA, International School of Advanced Studies, Trieste, Italy
YouTube : https://youtu.be/TClWpXH_pZg
Organizers : Pavan Pranjivan Mehta* (pavan.mehta@sissa.it) and Arran Fernandez** (arran.fernandez@emu.edu.tr)
* SISSA, International School of Advanced Studies, Italy
** Eastern Mediterranean University, Northern Cyprus
Keywords: integral fractional Laplacian, regularity, singularity structure, low-order terms
Abstract: We study certain fractional elliptic equations that include lower-order terms. It is well-known that the fractional Laplacian introduces a mild singularity near the boundary of the bounded domain. Despite this, adding lower-order terms does not reduce the solution's smoothness in Hölder and Sobolev spaces, either near the boundary or within the domain. However, numerical methods can suffer in accuracy. This is particularly the case when the weak boundary singularity is explicitly incorporated into the computed solution. We demonstrate that this occurs with spectral methods that use weighted Jacobi polynomials for computations on an interval. We also find similar effects for spectral Galerkin methods on a disk. Additionally, we conjecture that the pattern of singularities in these solutions is similar to the structure in Mittag-Leffler functions.
Biography: Zhongqiang Zhang (张中强) is an Associate Professor of Mathematics at Worcester Polytechnic Institute. His research interests include numerical methods for stochastic and integral differential equations, computational probability, and mathematics for machine learning. Before he joined in Worcester Polytechnic Institute in 2014, he received Ph.D. degrees in mathematics at Shanghai University in 2011 and in applied mathematics at Brown University in 2014. He co-authored a book with George Karniadakis on numerical methods for stochastic partial differential equations with white noise.
Bibliography
[1] Z. Hao and Z. Zhang. Optimal regularity and error estimates of a spectral Galerkin method for fractional advection-diffusion-reaction equations. SIAM J. Numer. Anal., 58(1):211–233, 2020
[2] Z. Hao and Z. Zhang. Numerical approximation of optimal convergence for fractional elliptic equations with additive fractional Gaussian noise. SIAM/ASA J. Uncertain. Quantif., 9(3):1013–1033, 2021
[3] Z. Hao, H. Li, Z. Zhang, and Z. Zhang. Sharp error estimates of a spectral Galerkin method for a diffusion-reaction equation with integral fractional Laplacian on a disk. Math. Comp., 90(331):2107–2135, 2021
[4] Q. Zhuang, A. Heryudono, F. Zeng, and Z. Zhang. Collocation methods for integral fractional Laplacian and fractional PDEs based on radial basis functions. Appl. Math. Comput., 469:Paper No. 128548, 14, 2024
[5] Z. Zhang. Error estimates of spectral Galerkin methods for a linear fractional reaction-diffusion equation. J. Sci. Comput., 78(2):1087–1110, 2019
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