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Bloch theory-based gradient recovery method for computing topological edge modes in photonic graphene

Hailong Guo, Xu Yang, and Yi Zhu
Submitted Paper

Abstract

Photonic graphene, a photonic crystal with honeycomb structures, has been intensively studied in both theoretical and applied fields. Similar to graphene which admits Dirac Fermions and topological edge states, photonic graphene supports novel and subtle propagating modes (edge modes) of electromagnetic waves. These modes have wide applications in many optical systems. In this paper, we propose a novel gradient recovery method based on Bloch theory for the computation of topological edge modes in the honeycomb structure. Compared to standard finite element methods, this method provides higher order accuracy with the help of gradient recovery technique. This high order accuracy is highly desired for constructing the propagating electromagnetic modes in applications. We analyze the accuracy and prove the superconvergence of this method. Numerical examples are presented to show the efficiency by computing the edge mode for the P-symmetry and C-symmetry breaking cases in honeycomb structures.

Gradient recovery for elliptic interface problem: III. Nitsche's method

Hailong Guo and Xu Yang
Submitted Paper

Abstract

This is the third paper on the study of gradient recovery for elliptic interface problem. In our previous works [H. Guo and X. Yang, 2016, arXiv:1607.05898 and {\it J. Comput. Phys.}, 338 (2017), 606--619], we developed {gradient recovery methods} for elliptic interface problem based on body-fitted meshes and immersed finite element methods. Despite the efficiency and accuracy that these methods bring to {recover} the gradient, there are still some cases in unfitted meshes where skinny triangles appear in the generated local body-fitted {triangulations} that destroy the accuracy of recovered gradient near the interface. In this paper, we propose a gradient recovery technique based on Nitsche's method for elliptic interface problem, which avoids the loss of accuracy of gradient near the interface caused by skinny triangles. We analyze the supercloseness between the gradient of the numerical solution by the Nitsche's method and the gradient of { the} interpolation of the exact solution, which leads to the superconvergence of the proposed gradient recovery method. We also present several numerical examples to validate the theoretical results.

Parametric polynomial preserving recovery on manifolds

Guozhi Dong and Hailong Guo
Submitted Paper

Abstract

This paper investigates gradient recovery schemes for data defined on discretized manifolds. The proposed method, parametric polynomial preserving recovery (PPPR), does not ask for the tangent spaces of the exact manifolds which have been assumed for some significant gradient recovery methods in the literature. Another advantage of the proposed method is that it removes the symmetric requirement from the existing methods for the superconvergence. These properties make it a prime method when meshes are arbitrarily structured or generated from high curvature surfaces. As an application, we show that the recovery operator is capable of constructing an asymptotically exact posteriori error estimator. Several numerical examples on 2--dimensional surfaces are presented to support the theoretical results and make comparisons with methods in the state of the art, which show evidence that the PPPR method outperforms the existing methods.

A $C^0$ Linear Finite Element Method For Sixth Order Elliptic Equations

Hailong Guo, Qingsong Zou, and Zhimin Zhang
Submitted Paper

Abstract

TBD

A $C^0$ Linear Finite Element Method for Biharmonic Problems

Hailong Guo, Qingsong Zou, and Zhimin Zhang
Accepted Paper J. Sci. Comput. (2017)

Abstract

In this paper, a $C^0$ linear finite element method for biharmonic equations is constructed and analyzed. In our construction, the popular post-processing gradient recovery operators are used to calculate approximately the second order partial derivatives of a $C^0$ linear finite element function which do not exist in traditional meaning. The proposed scheme is straightforward and simple. More importantly, it is shown that the numerical solution of the proposed method converges to the exact one with optimal orders both under $L^2$ and discrete $H^2$ norms, while the recovered numerical gradient converges to the exact one with superconvergence order. Some novel properties of gradient recovery operators are discovered in the analysis of our method. In several numerical experiments, our theoretical findings are verified and a comparison of the proposed method with the nonconforming Morley element and $C^0$ interior penalty method is given.

Superconvergence analysis of partially penalized immersed finite element methods

Hailong Guo, Zhimin Zhang, and Xu Yang
Accepted Paper IMA J. Numer. Anal. (2017).

Abstract

The contribution of this article contains two parts: first, we prove a supercloseness result for the partially penalized immersed finite element (PPIFE) methods in (Lin, T., Lin, Y. & Zhang, X. (2015), Partially penalized immersed finite element methods for elliptic interface problems. SIAM J. Numer. Anal., 53, 1121–1144) and then based on the supercloseness result, we show that the gradient recovery method proposed in our previous work (Guo, H. & Yang, X. (2017) Gradient recovery for elliptic interface problem: II. Immersed finite element methods. J. Comput. Phys., 338, 606–619) can be applied to the PPIFE methods and the recovered gradient converges to the exact gradient with a superconvergent rate $O(h^{1.5})$. Hence, the gradient recovery method provides an asymptotically exact a posteriori error estimator for the PPIFE methods. Several numerical examples are presented to verify our theoretical results.

Gradient recovery for elliptic interface problem: I. body-fitted mesh

Hailong Guo, and Xu Yang
Accepted Paper Commun. Comput. Phys. (2017).

Abstract

In this paper, we propose a novel gradient recovery method for elliptic interface problem using body-fitted mesh in two dimension. Due to the lack of regularity of solution at interface, standard gradient recovery methods fail to give superconvergent results, and thus will lead to overrefinement when served as a posteriori error estimator. This drawback is overcome by designing an immersed gradient recovery operator in our method. We prove the superconvergence of this method for both mildly unstructured mesh and adaptive mesh, and present several numerical examples to verify the superconvergence and its robustness as a posteriori error estimator.

Gradient recovery for elliptic interface problem: II. immersed finite element methods

Hailong Guo, and Xu Yang
Journal Paper J. Comput. Phys. 338 (2017), 606--619.

Abstract

This is the second paper on the study of gradient recovery for elliptic interface problem. In our previous work [H. Guo and X. Yang, 2016, arXiv:1607.05898], we developed gradient recovery finite element method based on body-fitted mesh. In this paper, we propose new gradient recovery methods based on two immersed interface finite element methods: symmetric and consistent immersed finite method [H. Ji, J. Chen and Z. Li, J. Sci. Comput., 61 (2014), 533--557] and Petrov-Galerkin immersed finite element method [T.Y. Hou, X.-H. Wu and Y. Zhang, Commun. Math. Sci., 2 (2004), 185--205, and S. Hou and X.-D. Liu, J. Comput. Phys., 202 (2005), 411--445]. Compared to body-fitted mesh based gradient recover methods, immersed finite element methods provide a uniform way of recovering gradient on regular meshes. Numerical examples are presented to confirm the superconvergence of both gradient recovery methods. Moreover, they provide asymptotically exact a posteriori error estimators for both immersed finite element methods.

A $C^0$ Linear Finite Element Method For Two Fourth-Order Eigenvalue Problems

Hongtao Chen, Hailong Guo, Zhimin Zhang and Qingsong Zou
Journal Paper IMA J. Numer. Anal., 37 (2017), 2120--2138.

Abstract

In this article, we construct a $C^0$ linear finite element method for two fourth-order eigenvalue problems: the biharmonic and the transmission eigenvalue problems. The basic idea of our construction is to use gradient recovery operator to compute the higher-order derivatives of a $C^0$ piecewise linear function, which do not exist in the classical sense. For the biharmonic eigenvalue problem, the optimal convergence rates of eigenvalue/eigenfunction approximation are theoretically derived and numerically verified. For the transmission eigenvalue problem, the optimal convergence rate of the eigenvalues is verified by two numerical examples: one for constant refraction index and the other for variable refraction index. Compared with existing schemes in the literature, the proposed scheme is straightforward and simpler, and computationally less expensive to achieve the same order of accuracy.

Polynomial Preserving Recovery For High Frequency Wave Propagation

Hailong Guo and Xu Yang
Journal Paper J. Sci. Comput. 71 (2017), no. 2, 594-614.

Abstract

Polynomial preserving recovery (PPR) was first proposed and analyzed in [Z. Zhang and A. Naga, {\it SIAM J. Sci. Comput.}, 26 (2005), 1192-1213], with intensive following applications on elliptic problems. In this paper, we generalize the study of PPR to high-frequency wave propagation. Specifically, we establish the supercloseness between finite element solution and its interpolation with explicit dependence on the frequency of wavefield, and then prove the superconvergence of PPR for high-frequency solutions to wave equation based on the supercloseness. We also present several numerical examples of PPR for both low-frequency and high-frequency wave propagation in order to confirm the theoretical results of superconvergence analysis.

Hessian Recovery For Finite Element Methods

Hailong Guo , Zhimin Zhang and Ren Zhao
Journal PaperMath. Comp. 86 (2017), no. 306, 1671--1692.

Abstract

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(hk) 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.

Superconvergent Two-Grid Methods For Elliptic Eigenvalue Problems

Hailong Guo, Zhimin Zhang and Ren Zhao
Journal Paper J. Sci. Comput. 70 (2017), no. 1, 125-148.

Abstract

Some numerical algorithms for elliptic eigenvalue problems are proposed, analyzed, and numerically tested. The methods combine advantages of the two-grid algorithm [J. Xu and A. Zhou, Math. Comp, 70(2001), 17–25], two-space method [M.R. Racheva and A. B. Andreev, Comput. Methods Appl. Math., 2(2002), 171–185], the shifted inverse power method [X. Hu and X. Cheng, Math. Comp., 80(2011), 1287–1301; Y. Yang and H. Bi, SIAM J. Numer. Anal, 49(2011), 1602–1624], and the polynomial preserving recovery technique [Z. Zhang and A. Naga, SIAM J. Sci. Comput., 26(2005), 1192–1213]. Our new algorithms compare favorably with some existing methods and enjoy superconvergence property.

Polynomial preserving recovery on boundary

Hailong Guo, Zhimin Zhang, Ren Zhao, and Qingsong Zou
Journal Paper J. Comput. Appl. Math. 307 (2016), 119--133.

Abstract

In this paper, we propose two systematic strategies to recover the gradient on the boundary of a domain. The recovered gradient has comparable superconvergent property on the boundary as that in the interior of the domain. This superconvergence property has been validated by several numerical experiments.

Gradient Recovery for the Crouzeix–Raviart Element

Hailong Guo and Zhimin Zhang
Journal Paper J. Sci. Comput. 64 (2015), no. 2, 456--476.

Abstract

A gradient recovery method for the Crouzeix–Raviart element is proposed and analyzed. The proposed method is based on local discrete least square fittings. It is proven to preserve quadratic polynomials and be a bounded linear operator. Numerical examples indicate that it can produce a superconvergent gradient approximation for both elliptic equations and Stokes equations. In addition, it provides an asymptotically exact posteriori error estimators for the Crouzeix–Raviart element.

Superconvergence of conforming finite element for fourth-order singularly perturbed problems of reaction diffusion type in 1D

Hailong Guo, Can Huang and Zhimin Zhang
Journal Paper Numer. Methods Partial Differential Equations 30 (2014), no. 2, 550--566.

Abstract

We consider conforming finite element approximation of fourth-order singularly perturbed problems of reaction diffusion type. We prove superconvergence of standard C1 finite element method of degree p on a modified Shishkin mesh. In particular, a superconvergence error bound of (N^{-1}\ln (N+1))^p in a discrete energy norm is established. The error bound is uniformly valid with respect to the singular perturbation parameter ϵ. Numerical tests indicate that the error estimate is sharp.

A spectral collocation method for eigenvalue problems of compact integral operators

Can Huang, Hailong Guo and Zhimin Zhang
Journal Paper J. Integral Equations Appl. 25 (2013), no. 1, 79--101.

Abstract

We propose and analyze a new spectral collo- cation method to solve eigenvalue problems of compact inte- gral operators, particularly, piecewise smooth operator kernels and weakly singular operator kernels of the form 1/|t − s| μ, 0 < μ < 1. We prove that the convergence rate of eigen- value approximation depends upon the smoothness of the cor- responding eigenfunctions for piecewise smooth kernels. On the other hand, we can numerically obtain a higher rate of convergence for the above weakly singular kernel for some μ’s even if the eigenfunction is not smooth. Numerical experi- ments confirm our theoretical results.