arXiv subjects

Zongze Yang

Publications and source records attributed to Zongze Yang.

8 recordsLinked to original sources

arXiv2026

A Second-order Structure-preserving Parametric FEM for Surface Evolution

In this paper, we propose a second-order-in-time, structure-preserving, and mesh-robust parametric finite element method for surface diffusion and volume-preserving mean curvature flow. We first reformulate the original evolution equations into new systems in which the tangential motion is governed by a harmonic map heat flow. This heat flow maps a fixed reference surface onto the unknown evolving surface and drives points on the evolving surface to move in their tangent spaces so as to reduce the associated harmonic energy. As a result, in the discrete setting, the mesh quality can be maintained at a level comparable to that of the reference surface, unless singularities occur. The volume-preserving property is theoretically guaranteed by the careful design of the scheme, while energy dissipation is enforced through a Lagrange multiplier. We present several numerical experiments to demonstrate second-order convergence in time and the advantage of the proposed method in preserving mesh quality. The structure-preserving properties are further confirmed by the numerical results. Finally, the proposed framework can be readily extended to other geometric flows.

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arXiv2025

Higher-order multiscale method and its convergence analysis for nonlinear thermo-electric coupling problems of composite structures

This paper proposes a higher-order multiscale computational method for nonlinear thermo-electric coupling problems of composite structures, which possess temperature-dependent material properties and nonlinear Joule heating. The innovative contributions of this work are the novel multiscale formulation with the higher-order correction terms for periodic composite structures and the global error estimation with an explicit rate for higher-order multiscale solutions. By employing the multiscale asymptotic approach and the Taylor series technique, the higher-order multiscale method is established for time-dependent nonlinear thermo-electric coupling problems, which can keep the local balance of heat flux and electric charge for high-accuracy multiscale simulation. Furthermore, an efficient numerical algorithm with off-line and on-line stages is presented in detail, and corresponding convergent analysis is also obtained. Two- and three-dimensional numerical experiments are conducted to showcase the competitive advantages of the proposed method for simulating the time-dependent nonlinear thermo-electric coupling problems in composite structures, not only exceptional numerical accuracy, but also less computational cost.

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arXiv2023

High-order splitting finite element methods for the subdiffusion equation with limited smoothing property

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 this paper, a new splitting of the solution is constructed for high-order finite element approximations to the subdiffusion equation with nonsmooth initial data. The method is constructed by splitting the solution into two parts, i.e., a time-dependent smooth part and a time-independent nonsmooth part, and then approximating the two parts via different strategies. The time-dependent smooth part is approximated by using high-order finite element method in space and convolution quadrature in time, while the steady nonsmooth part could be approximated by using smaller mesh size or other methods that could yield high-order accuracy. Several examples are presented to show how to accurately approximate the steady nonsmooth part, including piecewise smooth initial data, Dirac--Delta point initial data, and Dirac measure concentrated on an interface. The argument could be directly extended to subdiffusion equations with nonsmooth source data. Extensive numerical experiments are presented to support the theoretical analysis and to illustrate the performance of the proposed high-order splitting finite element methods.

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arXiv2023

A quadrature scheme for steady-state diffusion equations involving fractional power of regularly accretive operator

In this paper, we construct a quadrature scheme to numerically solve the nonlocal diffusion equation $(\mathcal{A}^α+b\mathcal{I})u=f$ with $\mathcal{A}^α$ the $α$-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. The error estimates include a wide range of cases in which the regularity index and spectral angle of $\mathcal{A}$, the smoothness of $f$, the size of $b$ and $α$ are all involved. The quadrature scheme is exponentially convergent with respect to the step size and is root-exponentially convergent with respect to the number of solves. Some numerical tests are presented in the last section to verify the sharpness of our estimates. Furthermore, both the scheme and the error bounds can be utilized directly to solve and analyze time-dependent problems.

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arXiv2019

Finite Element Methods for Fractional PDEs in Three Dimensions

This paper is a generalization of the previous work (Yang et.al, J. Comput. Phys. 330 (2017), 863-883) to the 3-D irregular convex domains. The analytical calculation formula of fractional derivatives of finite element basis functions are given and a path searching method is developed to find the integration paths corresponding to the Gaussian points. Moreover, a template matrix is introduced to speed up the procedures. Numerical experiments on a steady problem are presented verifying the efficiency of proposed techniques.

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arXiv2017

Numerical algorithm for two-dimensional time-fractional wave equation of distributed-order with a nonlinear source term

In this paper, an alternating direction implicit (ADI) difference scheme for two-dimensional time-fractional wave equation of distributed-order with a nonlinear source term is presented. The unique solvability of the difference solution is discussed, and the unconditional stability and convergence order of the numerical scheme are analysed. Finally, numerical experiments are carried out to verify the effectiveness and accuracy of the algorithm.

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arXiv2016

Maximum-norm error analysis of compact difference schemes for the backward fractional Feynman-Kac equation

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 discretized schemes for fractional substantial derivatives proposed recently, we construct compact finite difference schemes for the backward fractional Feynman-Kac equation, which has q-th (q=1, 2, 3, 4) order accuracy in temporal direction and fourth order accuracy in spatial direction, respectively. In the case q=1, the numerical stability and convergence of the difference scheme in the discrete L-infinity norm are proved strictly, where a new inner product is defined for the theoretical analysis. Finally, numerical examples are provided to verify the effectiveness and accuracy of the algorithms.

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arXiv2016

Effective numerical treatment of sub-diffusion equation with non-smooth solution

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 Nystrom method in temporal direction with the compact finite difference method and the spectral method in spatial direction. The distinct advantage of this approach in comparison with most current methods is its high convergence rate even though the solution of the anomalous sub-diffusion equation usually has lower regularity on the starting point. The effectiveness and efficiency of our proposed method are verified by several numerical experiments.

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