Waixiang Cao

Waixiang Cao, Xu Zhang, Zhimin Zhang

In this article, we study superconvergence properties of immersed finite element methods for the one dimensional elliptic interface problem. Due to low global regularity of the solution, classical superconvergence phenomenon for finite element methods disappears unless the discontinuity of the coefficient is resolved by partition. We show that immersed finite element solutions inherit all desired superconvergence properties from standard finite element methods without requiring the mesh to be aligned with the interface. In particular, on interface elements, superconvergence occurs at roots of generalized orthogonal polynomials that satisfy both orthogonality and interface jump conditions.

Lizhen Chen, Jia Zhao, Waixiang Cao et al.

In this paper, we propose a time-fractional molecular beam epitaxy (MBE) model with slope selection and its efficient, accurate, full discrete, linear numerical approximation. The numerical scheme utilizes the fast algorithm for the Caputo fractional derivative operator in time discretization and Fourier spectral method in spatial discretization. Refinement tests are conducted to verify the $2-α$ order of time convergence, with $α\in (0, 1]$ the fractional order of derivative. Several numerical simulations are presented to demonstrate the accuracy and efficiency of our newly proposed scheme. By exploring the fast algorithm calculating the Caputo fractional derivative, our numerical scheme makes it practice for long time simulation of MBE coarsening, which is essential for MBE model in practice. With the proposed fractional MBE model, we observe that the scaling law for the energy decays as $ O(t^{-\fracα{3}})$ and the roughness increases as $O(t^{\fracα{3}})$, during the coarsening dynamics with random initial condition. That is to say, the coarsening rate of MBE model could be manipulated by the fractional order $α$, and it is linearly proportional to $α$. This is the first time in literature to report/discover such scaling correlation. It provides a potential application field for fractional differential equations. Besides, the numerical approximation strategy proposed in this paper can be readily applied to study many classes of time-fractional and high dimensional phase field models.

Waixiang Cao, Zhimin Zhang, Qingsong Zou

We present and analyze a finite volume scheme of arbitrary order for elliptic equations in the one-dimensional setting. In this scheme, the control volumes are constructed by using the Gauss points in subintervals of the underlying mesh. We provide a unified proof for the inf-sup condition, and show that our finite volume scheme has optimal convergence rate under the energy and $L^2$ norms of the approximate error. Furthermore, we prove that the derivative error is superconvergent at all Gauss points and in some special case, the convergence rate can reach $h^{2r}$, where $r$ is the polynomial degree of the trial space. All theoretical results are justified by numerical tests.

Waixiang Cao, Xu Zhang, Zhimin Zhang et al.

In this paper, we introduce a class of high order immersed finite volume methods (IFVM) for one-dimensional interface problems. We show the optimal convergence of IFVM in H1 and L2 norms. We also prove some superconvergence results of IFVM. To be more precise, the IFVM solution is superconvergent of order p+2 at the roots of generalized Lobatto polynomials, and the flux is superconvergent of order p+1 at generalized Gauss points on each element including the interface element. Furthermore, for diffusion interface problems, the convergence rates for IFVM solution at the mesh points and the flux at generalized Gaussian points can both be raised to 2p. These superconvergence results are consistent with those for the standard finite volume methods. Numerical examples are provided to confirm our theoretical analysis.