arXiv subjects

Zhimin Zhang

Publications and source records attributed to Zhimin Zhang.

At least 109 records · Page 6Linked to original sources

arXiv2016

Spectral-Galerkin Approximation and Optimal Error Estimate for Stokes Eigenvalue Problems in Polar Geometries

In this paper we propose and analyze spectral-Galerkin methods for the Stokes eigenvalue problem based on the stream function formulation in polar geometries. We first analyze the stream function} formulated fourth-order equation under the polar coordinates, then we derive the pole condition and reduce the problem on a circular disk to a sequence of equivalent one-dimensional eigenvalue problems that can be solved in parallel. The novelty of our approach lies in the construction} of suitably weighted Sobolev spaces according to the pole conditions, based on which, the optimal error estimate for approximated eigenvalue of each one dimensional problem can be obtained. Further, we extend our method to the non-separable Stokes eigenvalue problem in an elliptic domain and establish the optimal error bounds. Finally, we provide some numerical experiments to validate our theoretical results and algorithms.

math.NA↗

arXiv2016

A Multilevel Correction Scheme for Nonsymmetric Eigenvalue Problems by Finite Element Methods

A multilevel correction scheme is proposed to solve defective and nodefective of nonsymmetric partial differential operators by the finite element method. The method includes multi correction steps in a sequence of finite element spaces. In each correction step, we only need to solve two source problems on a finer finite element space and two eigenvalue problems on the coarsest finite element space. The accuracy of the eigenpair approximation is improved after each correction step. This correction scheme improves overall efficiency of the finite element method in solving nonsymmetric eigenvalue problems.

math.NA↗

arXiv2016

Robust residual-based a posteriori error estimators for mixed finite element methods for fourth order elliptic singularly perturbed problems

We consider mixed finite element approximation of a singularly perturbed fourth-order elliptic problem with two different boundary conditions, and present a new measure of the error, whose components are balanced with respect to the perturbation parameter. Robust residual-based a posteriori estimators for the new measure are obtained, which are achieved via a novel analytical technique based on an approximation result. Numerical examples are presented to validate our theory.

math.NA↗

arXiv2016

Robust recovery-type a posteriori error estimators for streamline upwind/Petrov Galerkin discretizations for singularly perturbed problems

In this paper, we investigate adaptive streamline upwind/Petrov Galerkin (SUPG) methods for singularly perturbed convection-diffusion-reaction equations in a new dual norm presented in [Du and Zhang, J. Sci. Comput. (2015)]. The flux is recovered by either local averaging in conforming $H({\rm div})$ spaces or weighted global $L^2$ projection onto conforming $H({\rm div})$ spaces. We further introduce a recovery stabilization procedure, and develop completely robust a posteriori error estimators with respect to the singular perturbation parameter $\varepsilon$. Numerical experiments are reported to support the theoretical results and to show that the estimated errors depend on the degrees of freedom uniformly in $\varepsilon$.

math.NA↗

arXiv2016

Efficient Spectral and Spectral Element Methods for Eigenvalue Problems of Schrödinger Equations with an Inverse Square Potential

In this article, we study numerical approximation of eigenvalue problems of the Schrödinger operator $\displaystyle -Δu + \frac{c^2}{|x|^2}u$. There are three stages in our investigation: We start from a ball of any dimension, in which case the exact solution in the radial direction can be expressed by Bessel functions of fractional degrees. This knowledge helps us to design two novel spectral methods by modifying the polynomial basis to fit the singularities of the eigenfunctions. At the second stage, we move to circular sectors in the two dimensional setting. Again the radial direction can be expressed by Bessel functions of fractional degrees. Only in the tangential direction some modifications are needed from stage one. At the final stage, we extend the idea to arbitrary polygonal domains. We propose a mortar spectral element approach: a polygonal domain is decomposed into several sub-domains with each singular corner including the origin covered by a circular sector, in which origin and corner singularities are handled similarly as in the former stages, and the remaining domains are either a standard quadrilateral/triangle or a quadrilateral/triangle with a circular edge, in which the traditional polynomial based spectral method is applied. All sub-domains are linked by mortar elements (note that we may have hanging nodes). In all three stages, exponential convergence rates are achieved. Numerical experiments indicate that our new methods are superior to standard polynomial based spectral (or spectral element) methods and $hp$-adaptive methods. Our study offers a new and effective way to handle eigenvalue problems of the Schrödinger operator including the Laplacian operator on polygonal domains with reentrant corners.

math.NA↗

arXiv2015

Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations

We present an efficient algorithm for the evaluation of the Caputo fractional derivative $_0^C\!D_t^αf(t)$ of order $α\in (0,1)$, 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 $[Δt, T]$ with a uniform absolute error $\varepsilon$, where the number of exponentials $N_{\text{exp}}$ needed is of the order $O\left(\log\frac{1}{\varepsilon}\left( \log\log\frac{1}{\varepsilon}+\log\frac{T}{Δt}\right) +\log\frac{1}{Δt}\left( \log\log\frac{1}{\varepsilon}+\log\frac{1}{Δt}\right) \right)$. As compared with the direct method, the resulting algorithm reduces the storage requirement from $O(N_T)$ to $O(N_{\text{exp}})$ and the overall computational cost from $O(N_T^2)$ to $O(N_TN_{\text{exp}})$ with $N_T$ the total number of time steps. Furthermore, when the fast evaluation scheme of the Caputo derivative is applied to solve the fractional diffusion equations, the resulting algorithm requires only $O(N_SN_{\text{exp}})$ storage and $O(N_SN_TN_{\text{exp}})$ work with $N_S$ the total number of points in space; whereas the direct methods require $O(N_SN_T$) storage and $O(N_SN_T^2)$ work. The complexity of both algorithms is nearly optimal since $N_{\text{exp}}$ is of the order $O(\log N_T)$ for $T\gg 1$ or $O(\log^2N_T)$ for $T\approx 1$ for fixed accuracy $\varepsilon$. We also present a detailed stability and error analysis of the new scheme for solving linear fractional diffusion equations. The performance of the new algorithm is illustrated via several numerical examples. Finally, the algorithm can be parallelized in a straightforward manner.

math.NA↗

arXiv2015

Dynamical Evolution of Anisotropic Response in Black Phosphorus under Ultrafast Photoexcitation

Black phosphorus has recently emerged as a promising material for high performance electronic and optoelectronic device for its high mobility, tunable mid-infrared bandgap and anisotropic electronic properties. Dynamical evolution of photo excited carriers and its induced change of transient electronic properties are critical for materials' high field performance, but remains to be explored for black phosphorus. In this work, we perform angle resolved transient reflection spectroscopy to study the dynamical evolution of anisotropic properties of black phosphorus under photo excitation. We find that the anisotropy of reflectivity is enhanced in the pump induced quasi-equilibrium state, suggesting an extraordinary enhancement of the anisotropy in dynamical conductivity in hot carrier dominated regime. These results raise enormous possibilities of creating high field, angle sensitive electronic, optoelectronic and remote sensing devices exploiting the dynamical electronic anisotropic with black phosphorus.

cond-mat.mtrl-sci↗

arXiv2015

Superconvergence points of fractional spectral interpolation

We investigate superconvergence properties of the spectral interpolation involving fractional derivatives. Our interest in this superconvergence problem is, in fact, twofold: when interpolating function values, we identify the points at which fractional derivatives of the interpolant superconverge; when interpolating fractional derivatives, we locate those points where function values of the interpolant superconverge. For the former case, we apply various Legendre polynomials as basis functions and obtain the superconvergence points, which naturally unify the superconvergence points for the first order derivative presented in [Z. Zhang, SIAM J. Numer. Anal., 50 (2012), 2966-2985], depending on orders of fractional derivatives. While for the latter case, we utilize Petrov-Galerkin method based on generalized Jacobi functions (GJF) [S. Chen et al., arXiv: 1407. 8303v1] and locate the superconvergence points both for function values and fractional derivatives. Numerical examples are provided to verify the analysis of superconvergence points for each case.

math.NA↗

arXiv2015

Sharp Error Bounds for Piecewise Polynomial Approximation: Revisit and Application to Elliptic PDE Eigenvalue Computation

In this paper, we revisit approximation properties of piecewise polynomial spaces, which contain more than ${\cal P}_{r-1}$ but not ${\cal P}_r$. We develop more accurate upper and lower error bounds that are sharper than those used in literature. These new error bounds, especially the lower bounds are particular useful to finite element methods. As an important application, we establish sharp lower bounds of the discretization error for Laplace and $2m$-th order elliptic eigenvalue problems in various finite element spaces under shape regular triangulations, and investigate the asymptotic convergence behavior for large numerical eigenvalue approximations.

math.NA↗

arXiv2014

Superconvergent Two-grid Methods For Elliptic Eigenvalue Problems

Some numerical algorithms for elliptic eigenvalue problems are proposed, analyzed, and numerically tested. The methods combine advantages of the two-grid algorithm, two-space method, the shifted inverse power method, and the polynomial preserving recovery technique . Our new algorithms compare favorably with some existing methods and enjoy superconvergence property.

math.NA↗

arXiv2014

Mathematical and numerical analysis of time-dependent Ginzburg--Landau equations in nonconvex polygons based on Hodge decomposition

We prove well-posedness of time-dependent Ginzburg--Landau system in a nonconvex polygonal domain, and decompose the solution as a regular part plus a singular part. We see that the magnetic potential is not in $H^1$ in general, and the finite element method (FEM) may give incorrect solutions. To remedy this situation, we reformulate the equations into an equivalent system of elliptic and parabolic equations based on the Hodge decomposition, which avoids direct calculation of the magnetic potential. The essential unknowns of the reformulated system admit $H^1$ solutions and can be solved correctly by the FEMs. We then propose a decoupled and linearized FEM to solve the reformulated equations and present error estimates based on proved regularity of the solution. Numerical examples are provided to support our theoretical analysis and show the efficiency of the method.

math.NA↗

arXiv2014

A new approach for numerical simulation of the time-dependent Ginzburg-Landau equations

We introduce a new approach for finite element simulations of the time-dependent Ginzburg-Landau equations (TDGL) in a general curved polygon, possibly with reentrant corners. Specifically, we reformulate the TDGL into an equivalent system of equations by decomposing the magnetic potential to the sum of its divergence-free and curl-free parts, respectively. Numerical simulations of vortex dynamics show that, in a domain with reentrant corners, the new approach is much more stable and accurate than the old approaches of solving the TDGL directly (under either the temporal gauge or the Lorentz gauge); in a convex domain, the new approach gives comparably accurate solutions as the old approaches.

math.NA↗

arXiv2014

Hessian Recovery for Finite Element Methods

In this article, we propose and analyze an effective Hessian recovery strategy for the Lagrangian finite element of arbitrary order $k$. We prove that the proposed Hessian recovery preserves polynomials of degree $k+1$ on general unstructured meshes and superconverges at rate $O(h^k)$ on mildly structured meshes. In addition, the method preserves polynomials of degree $k+2$ on translation invariant meshes and produces a symmetric Hessian matrix when the sampling points for recovery are selected with symmetry. Numerical examples are presented to support our theoretical results.

math.NA↗

arXiv2014

Spectral method for substantial fractional differential equations

In this paper, a non-polynomial spectral Petrov-Galerkin method and associated collocation method for substantial fractional differential equations (FDEs) are proposed, analyzed, and tested. We extend a class of generalized Laguerre polynomials to form our basis. By a proper scaling of trial basis and test basis, our Petrov-Galerkin method results in a diagonal and thus well-conditioned linear systems for both fractional advection equation and fractional diffusion equation. In the meantime, we construct substantial fractional differential collocation matrices and provide explicit forms for both type of equations. Moreover, the proposed method allows us to adjust a parameter in basis selection according to different given data to maximize the convergence rate. This fact has been proved in our error analysis and confirmed in our numerical experiments.

math.NA↗

arXiv2014

Superconvergence of Local Discontinuous Galerkin method for one-dimensional linear parabolic equations

In this paper, we study superconvergence properties of the local discontinuous Galerkin method for one-dimensional linear parabolic equations when alternating fluxes are used. We prove, for any polynomial degree $k$, that the numerical fluxes converge at a rate of $2k+1$ (or $2k+1/2$) for all mesh nodes and the domain average under some suitable initial discretization. We further prove a $k+1$th superconvergence rate for the derivative approximation and a $k+2$th superconvergence rate for the function value approximation at the Radau points. Numerical experiments demonstrate that in most cases, our error estimates are optimal, i.e., the error bounds are sharp.

math.NA↗

arXiv2014

Is $2k$-Conjecture valid for finite volume methods?

This paper is concerned with superconvergence properties of a class of finite volume methods of arbitrary order over rectangular meshes. Our main result is to prove {\it 2k-conjecture}: at each vertex of the underlying rectangular mesh, the bi-$k$ degree finite volume solution approximates the exact solution with an order $ O(h^{2k})$, where $h$ is the mesh size. As byproducts, superconvergence properties for finite volume discretization errors at Lobatto and Gauss points are also obtained. All theoretical findings are confirmed by numerical experiments.

math.NA↗

arXiv2013

How Many Numerical Eigenvalues can We Trust?

When using finite element and finite difference methods to approximate eigenvalues of $2m^{th}$-order elliptic problems, the number of reliable numerical eigenvalues can be estimated in terms of the total degrees of freedom $N$ in resulting discrete systems. The truth is worse than what we used to believe in that the percentage of reliable eigenvalues decreases with an increased $N$, even though the number of reliable eigenvalues increases with $N$.

math.NA↗

arXiv2013

On $hp$-Convergence of PSWFs and A New Well-Conditioned Prolate-Collocation Scheme

The first purpose of this paper is to provide a rigorous proof for the nonconvergence of $h$-refinement in $hp$-approximation by the PSWFs, a surprising convergence property that was first observed by Boyd et al [J. Sci. Comput., 2013]. The second purpose is to offer a new basis that leads to spectral-collocation systems with condition numbers independent of $(c,N),$ the intrinsic bandwidth parameter and the number of collocation points. In addition, this work gives insights into the development of effective spectral algorithms using this non-polynomial basis. We in particular highlight that the collocation scheme together with a very practical rule for pairing up $(c,N)$ significantly outperforms the Legendre polynomial-based method (and likewise other Jacobi polynomial-based method) in approximating highly oscillatory bandlimited functions.

math.NA↗