Zongze Yang

The Hong Kong Polytechnic University | PolyU · Department of Applied Mathematics

In contrast with the diffusion equation which smoothens the initial data to C ∞ C^\infty for t > 0 t>0 (away from the corners/edges of the domain), the subdiffusion equation only exhibits limited spatial regularity. As a result, one generally cannot expect high-order accuracy in space in solving the subdiffusion equation with nonsmooth initial data...

In the perfect conductivity problem (i.e., the conductivity problem with perfectly conducting inclusions), the gradient of the electric field is often very large in a narrow region between two inclusions and blows up as the distance between the inclusions tends to zero. The rigorous error analysis for the computation of such perfect conductivity pr...

In contrast with the diffusion equation which smoothens the initial data to $C^\infty$ for $t>0$ (away from the corners/edges of the domain), the subdiffusion equation only exhibits limited spatial regularity. As a result, one generally cannot expect high-order accuracy in space in solving the subdiffusion equation with nonsmooth initial data. In t...

The paper is concerned with the analysis of a popular convex-splitting finite element method for the Cahn-Hilliard-Navier-Stokes system, which has been widely used in practice. Since the method is based on a combined approximation to multiple variables involved in the system, the approximation to one of the variables may seriously affect the accura...

In this paper, we construct a quadrature scheme to numerically solve the nonlocal diffusion equation $(\mathcal{A}^\alpha+b\mathcal{I})u=f$ with $\mathcal{A}^\alpha$ the $\alpha$-th power of the regularly accretive operator $\mathcal{A}$. Rigorous error analysis is carried out and sharp error bounds (up to some negligible constants) are obtained. T...

We propose a linearized semi-implicit and decoupled finite element method for the incompressible Navier–Stokes equations with variable density. Our method is fully discrete and shown to be unconditionally stable. The velocity equation is solved by an H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \us...

A linearized fully discrete arbitrary Lagrangian–Eulerian finite element method is proposed for solving the two-phase Navier–Stokes flow system and to preserve the energy-diminishing structure of the system at the discrete level, by taking account of the kinetic, potential and surface energy. Two benchmark problems of rising bubbles in fluids in bo...

Though the finite element method has been widely used in solving fractional differential equations, the effects of the Gaussian quadrature rule on the numerical results have rarely been considered. Since the fractional derivatives of the basis functions are not polynomials with integer power and always have weak singularities on some elements, the...

A fully discrete surface finite element method is proposed for solving the viscous shallow water equations in a bounded Lipschitz domain on the sphere based on a general triangular mesh. The method consists of a modified Crank–Nicolson method in time and a Galerkin surface finite element method in space for the fluid thickness H and the fluid veloc...

An optimal-order error estimate is presented for the arbitrary Lagrangian-Eulerian (ALE) finite element method for a parabolic equation in an evolving domain, using high-order iso-parametric finite elements with flat simplices in the interior of the domain. The mesh velocity can be a linear approximation of a given bulk velocity field or a numerica...

In this paper, we use the finite element method (FEM) to solve the time-space fractional Bloch-Torrey equation on irregular domains in R3. Based on linear Lagrange basis functions, a space semi-discrete FEM scheme is given. By adopting the L2−1σ approximation for the Caputo fractional derivative, a fully discrete scheme is presented. Furthermore, w...

This paper is a generalization of the previous work (Yang et al., 2017) to the 3-D irregular convex domains. The analytical calculation formula of fractional derivatives of finite element basis functions is given and a path searching method is developed to find the integration paths corresponding to the Gaussian points. Moreover, a template matrix...

In quantum physics, fractional Schrödinger equation is of particular interest in the research of particles on stochastic fields modeled by the Lévy processes, which was derived by extending the Feynman path integral over the Brownian paths to a path integral over the trajectories of Lévy fights. In this work, a fully discrete finite element method...

In this paper, we address the main challenges of the implementation of finite element methods for solving spatial fractional problems on three dimensional irregular convex regions. Different from the integer case, the non-locality of fractional derivative operators makes the assembly of fractional stiffness matrix much more difficult, mainly in two...

Abstract Fractional differential equations (FDEs) of distributed-order are important in depicting the models where the order of differentiation distributes over a certain range. Numerically solving this kind of FDEs requires not only discretizations of the temporal and spatial derivatives, but also approximation of the distributed-order integral, w...

In this paper we investigate a sub-diffusion equation for simulating the anomalous diffusion phenomenon in real physical environment. Based on an equivalent transformation of the original sub-diffusion equation followed by the use of a smooth operator, we devise a high-order numerical scheme by combining the Nyström method in temporal direction wit...

In this article, we propose an exponential B-spline approach to obtain approximate solutions for the fractional sub-diffusion equation of Caputo type. The presented method is established via a uniform nodal collocation strategy by using an exponential B-spline based interpolation in conjunction with an effective finite difference scheme in time. Th...

In this article, we develop a fully discrete finite element method for the nonlinear Schrodinger equation (NLS) with time- and space-fractional derivatives. The time-fractional derivative is described in Caputo's sense and the space-fractional derivative in Riesz's sense. Its stability is well derived; the convergent estimate is discussed by an ort...

Space fractional advection diffusion equations are better to describe anomalous diffusion phenomena because of non-locality of fractional derivatives, which causes people to confront great trouble in problem solving while enjoying the convenience from mathematical modelling, especially in high dimensional cases. In this paper, we solve the three-di...

The fractional Feynman-Kac equations describe the distribution of functionals of non-Brownian motion, or anomalous diffusion, including two types called the forward and backward fractional Feynman-Kac equations, where the fractional substantial derivative is involved. This paper focuses on the more widely used backward version. Based on the discret...

In this paper we are concerned with anomalous sub-diffusion equation, which describes the anomalous diffusion phenomenon in nature. Based on an equivalent transformation of equation and a smoothing transformation, we propose high order schemes combining Nystrom method in temporal direction with (compact) finite difference method in spatial directio...

In this paper, we consider two dimensional Riesz space fractional diffusion equations with nonlinear source term on convex domain. In existing literatures, they only solved the equations on regular domain discretized by uniform structured meshes. Our breakthrough is developing an algorithm to form stiffness matrix on unstructured triangular meshes,...