Zhimin Zhang

Wayne State University | WSU · Department of Mathematics

Vibrations of structures subjected to concentrated point loads have many applications in mechanical engineering. Experiments are expensive and numerical methods are often used for simulations. In this paper, we consider the plate vibration with nonlinear dependence on the eigen-parameter. The problem is formulated as the eigenvalue problem of a hol...

The two-step backward differential formula (BDF2) implicit method with unequal time-steps is investigated for the Cahn-Hilliard model by focusing on the numerical influences of time-step variations. The suggested method is proved to preserve a modified energy dissipation law at the discrete levels if the adjoint time-step ratios fulfill a new step-...

Due to the intrinsically initial singularity of solution and the discrete convolution form in numerical Caputo derivatives, the traditional H 1-norm analysis (corresponding to the case for a classical diffusion equation) to the time approximations of a fractional subdiffusion problem always leads to suboptimal error estimates (a loss of time accura...

The variable two-step backward differentiation formula (BDF2) is revisited via a new theoretical framework using the positive semi-definiteness of BDF2 convolution kernels and a class of orthogonal convolution kernels. We prove that, if the adjacent time-step ratios $r_k:=\tau_k/\tau_{k-1}\le\brat{3+\sqrt{17}}/2\approx3.561$, the adaptive BDF2 time...

In this paper, we consider superconvergence error estimates of finite element method approximation of Caputo’s time fractional reaction–subdiffusion equations under nonuniform time meshes. For the standard Galerkin method we see that the optimal order error estimate of temporal direction cannot be derived from the weak formulation of the problem. W...

In this paper, we propose a new family of H(curl^2)-conforming elements for the quad-curl eigenvalue problem in 2D. The accuracy of this family is one order higher than that in [32]. We prove a priori and a posteriori error estimates. The a priori estimate of the eigenvalue with a convergence order 2(s-1) is obtained if the eigenvector u\in H^{s+1}...

The variable two-step backward differentiation formula (BDF2) is revisited via a new theoretical framework using the positive semi-definiteness of BDF2 convolution kernels and a class of orthogonal convolution kernels. We prove that, if the adjacent time-step ratios $r_k:=\tau_k/\tau_{k-1}\le(3+\sqrt{17})/2\approx3.561$, the adaptive BDF2 time-step...

The quad-curl problem arises from the inverse electromagnetic scattering theory and magnetohydrodynamics. In this paper, a weak Galerkin method is proposed using the curl-conforming Nédélec elements. On one hand, the method avoids the construction of the curl–curl conforming elements and thus solves a smaller linear system. On the other hand, it is...

Due to the intrinsically initial singularity of solution and the discrete convolution form in numerical Caputo derivatives, the traditional $H^1$-norm analysis (corresponding to the case for a classical diffusion equation) to the time approximations of a fractional subdiffusion problem always leads to suboptimal error estimates (a loss of time accu...

In this paper, we study convergence and superconvergence theory of integer and fractional derivatives of the one-point and the two-point Hermite interpolations. When considering the integer-order derivative, exponential decay of the error is proved, and superconvergence points are located, at which the convergence rates are $O(N^{-2})$ and $O(N^{-1...

An efficient numerical scheme is developed to solve a linearized time fractional KdV equation on unbounded spatial domains. First, the exact absorbing boundary conditions (ABCs) are derived which reduces the pure initial value problem into an equivalent initial-boundary value problem on a finite interval that contains the compact support of the ini...

We present an efficient algorithm for the evaluation of the Caputo fractional derivative C 0 D α t f (t) of order α ∈ (0, 1), which can be expressed as a convolution of f ′ (t) with the kernel t −α. The algorithm is based on an efficient sum-of-exponentials approximation for the kernel t −1−α on the interval [∆t, T ] with a uniform absolute error ε...

A polynomial preserving recovery technique is applied to an overpenalized symmetric interior penalty method. The discontinuous Galerkin solution values are used to recover the gradient and to further construct an a posteriori error estimator in the energy norm. In addition, for uniform triangular meshes and mildly structured meshes satisfying the ε...

This paper is concerned with superconvergence properties of discontinuous Galerkin (DG) methods for two-dimensional linear hyperbolic conservation laws over rectangular meshes when upwind fluxes are used. We prove, under some suitable initial and boundary discretizations, the (2k + 1)th order superconvergence rate of the DG approximation at the dow...

In this paper, we study superconvergence properties of the discontinuous Galerkin (DG) method for one-dimensional linear hyperbolic equation when upwind fluxes are used. We prove, for any polynomial degree $k$, the $2k+1$th (or $2k+1/2$th) superconvergence rate of the DG approximation at the downwind points and for the domain average under quasi-un...

In this paper, we consider discontinuous Galerkin approximations to the solution of Naghdi arches and show how to post-process them in an element-by-element fashion to obtain a far better approximation. Indeed, we prove that, if polynomials of degree k are used, the postprocessed approximation converges with order 2k+1 in the L 2 -norm throughout t...

In this article, an equilibrated gradient recovery error estimator is introduced and analyzed. Regional and global error bounds are established under the equilibrium condition. Furthermore, the error estimator based on the ZZ patch recovery technique is analyzed theoretically. Stability and consistent properties are proved under mild assumptions. A...

. Both nonconforming and enhanced strain methods are analyzed under the framework of the mixed method. The notion of selective nonconforming or selective enhanced strain methods are introduced. 1. Introduction. LetOmega ae R 2 be a polygonal domain. Consider onOmega , the plain strain problem in which we seek for the unknown displacement u 2 V such...

This is the first in a series of papers in which we discuss a numerical method to solve a certain class of steady state free boundary problems. The method constructs a sequence of solutions for the Laplace equation in different domains. We show in this paper that the sequence converges to the solution of given free boundary problem under the assump...

Convergence of cubic spline interpolation for discontinuous functions are investigated. It is shown that the complete cubic spline interpolation converges to the Heaviside step function in the L p -norm at rate O(h 1=p ) for quasi-uniform meshes when 1 p ! 1, and diverges in the L 1 -norm when the uniform meshes are used. No matter how small the me...

We present analytical and numerical investigation in the relationship of the recovery error estimator and the implicit residual error estimator. It is shown that analytically both error estimators are equivalent for one dimensional problems. Numerical study indicates that such equivalence also exist for two dimensional problems.

. In this work, the bilinear finite element method on a Shishkin mesh for convection-diffusion problems is analyzed in the two-dimensional setting. A superconvergent rate N Gamma2 ln 2 N +N Gamma3=2 is established on a discrete energy norm. This rate is uniformly valid with respect to the singular perturbation parameter ffl. As a by-product, an ffl...

A projected-shear finite element method for periodic Reissner–Mindlin plate model are analyzed for rectangular meshes. A projection operator is applied to the shear stress term in the bilinear form. Optimal error estimates in the L2-norm, the H1-norm, and the energy norm for both displacement and rotations are established and gradient superconverge...

Convergence of cubic spline interpolation for discontinuous functions are investigated. It is shown that the complete cubic spline interpolation converges to the Heaviside step function in the L p -norm at rate O(h 1=p ) for quasi-uniform meshes when 1 p ! 1, and diverges in the L 1 -norm when the uniform meshes are used. No matter how small the me...

Zienkiewicz-Zhu's derivative patch recovery technique is analyzed for general quadrilateral finite elements. Under certain regular conditions on the meshes, the arithmetic mean of the absolute error of the recovered gradient at the nodal points is superconvergent for the second-order elliptic operators. For rectangular meshes and the Laplacian, the...