Jiashun Hu

A novel evolving surface finite element method, based on a novel equivalent formulation of the continuous problem, is proposed for computing the evolution of a closed hypersurface moving under a prescribed velocity field in two- and threedimensional spaces. The method improves the mesh quality of the approximate surface by minimizing the rate of deformation using an artificial tangential motion. The transport evolution equations of the normal vector and the extrinsic Weingarten matrix are derived and coupled with the surface evolution equations to ensure stability and convergence of the numerical approximations. Optimal-order convergence of the semi-discrete evolving surface finite element method is proved for finite elements of degree k ≥ 2. Numerical examples are provided to illustrate the convergence of the proposed method and its effectiveness in improving mesh quality on the approximate evolving surface.

A class of high-order mass- and energy-conserving methods is proposed for the nonlinear Schr\o"dinger equation based on Gauss collocation in time and finite element discretization in space, by introducing a mass- and energy-correction post-process at every time level. The existence, uniqueness, and high-order convergence of the numerical solutions are proved. In particular, the error of the numerical solution is O(τk+1 + hp) in the L∞(0, T ; H1) norm after incorporating the accumulation errors arising from the post-processing correction procedure at all time levels, where k and p denote the degrees of finite elements in time and space, respectively, which can be arbitrarily large. Several numerical examples are provided to illustrate the performance of the proposed new method, including the conservation of mass and energy and the high-order convergence in simulating solitons and bi-solitons.

An artificial tangential velocity is introduced into the evolving finite element methods for mean curvature flow and Willmore flow proposed by Kovács et al. (Numer Math 143(4), 797-853, 2019, Numer Math 149, 595-643, 2021) in order to improve the mesh quality in the computation. The artificial tangential velocity is constructed by considering a limiting situation in the method proposed by Barrett et al. (J Comput Phys 222(1), 441-467, 2007, J Comput Phys 227(9), 4281-4307, 2008, SIAM J Sci Comput 31(1), 225-253, 2008) . The stability of the artificial tangential velocity is proved. The optimal-order convergence of the evolving finite element methods with artificial tangential velocity are proved for both mean curvature flow and Willmore flow. Extensive numerical experiments are presented to illustrate the convergence of the method and the performance of the artificial tangential velocity in improving the mesh quality.

A new type of bound states induced by PT symmetry is found which does not radiate but can be excited from the far field.

Numerical computation of a nonlocal diffusion equation on the real axis is considered in this paper. We first apply an extensively studied quadrature scheme to obtain a discrete nonlocal diffusion system on an unbounded domain. Then we derive an alternative formulation of the discrete problem based on the spectral analysis of the \z\-transform. This new formulation can be seen as a system defined on a bounded domain with an artificial boundary condition, and it allows us to reformulate the original infinite domain problem into an equivalent bounded domain problem. To numerically implement the exact artificial boundary condition, we apply the trapezoidal quadrature rule to approximate the contour integral induced by the inverse \z\-transform. Numerical examples are presented to demonstrate the effectiveness of our approach.