汇报标题 (Title)：Efficient spectral-Galerkin methods for PDEs in three-dimensional complex geometries（有效的谱Galerkin步骤求解三维复杂区域偏微分方程）
汇报人 (Speaker)：王中庆 教授（上海理工大学）
汇报功夫 (Time)：2023年11月17日(周五) 13:00-15:30
汇报地址 (Place)：腾讯会议：968-592-891
约请人(Inviter)：吴华
主办部门：理学院数学系
汇报提要：In this paper, we introduce a new spherical coordinate transformation, which transforms three-dimensional curved geometries into a unit sphere. This transformation plays an important role in spectral approximations of differential equations in three-dimensional curved geometries. Some basic properties of the spherical coordinate transformation are given. As examples, we consider an elliptic equation in three-dimensional curved geometries, prove the existence and uniqueness of the weak solution, construct the Fourier-Legendre spectral-Galerkin scheme and analyze the optimal convergence of numerical solutions under $H^1$-norm. We also apply the suggested approach to the Gross–Pitaevskii equation in three-dimensional curved geometries and present some numerical results. The proposed algorithm is very effective and easy to implement for problems in three-dimensional curved geometries. Abundant numerical results show that our spectral-Galerkin method possesses high order accuracy.
