Xiaoping Xie

Guozhu Yu, Xiaoping Xie, Carsten Carstensen

Assumed stress hybrid methods are known to improve the performance of standard displacement-based finite elements and are widely used in computational mechanics. The methods are based on the Hellinger-Reissner variational principle for the displacement and stress variables. This work analyzes two existing 4-node hybrid stress quadrilateral elements due to Pian and Sumihara [Int. J. Numer. Meth. Engng, 1984] and due to Xie and Zhou [Int. J. Numer. Meth. Engng, 2004], which behave robustly in numerical benchmark tests. For the finite elements, the isoparametric bilinear interpolation is used for the displacement approximation, while different piecewise-independent 5-parameter modes are employed for the stress approximation. We show that the two schemes are free from Poisson-locking, in the sense that the error bound in the a priori estimate is independent of the relevant Lame constant $λ$. We also establish the equivalence of the methods to two assumed enhanced strain schemes. Finally, we derive reliable and efficient residual-based a posteriori error estimators for the stress in $L^{2}$-norm and the displacement in $H^{1}$-norm, and verify the theoretical results by some numerical experiments.

Binjie Li, Hao Luo, Xiaoping Xie

This paper establishes the convergence of a time-steeping scheme for time fractional diffusion problems with nonsmooth data. We first analyze the regularity of the model problem with nonsmooth data, and then prove that the time-steeping scheme possesses optimal convergence rates in $ L^2(0,T;L^2(Ω)) $-norm and $ L^2(0,T;H_0^1(Ω)) $-norm with respect to the regularity of the solution. Finally, numerical results are provided to verify the theoretical results.

Li Wang, Xiaoping Xie

This paper analyzes rectangular finite element methods for fourth order elliptic singular perturbation problems. We show that the non-$C^0$ rectangular Morley element is uniformly convergent in the energy norm with respect to the perturbation parameter. We also propose a $C^0$ extended high order rectangular Morley element and prove the uniform convergence. Finally, we do some numerical experiments to confirm the theoretical results.

Binjie Li, Xiaoping Xie

In this paper, we analyze a family of hybridizable discontinuous Galerkin (HDG) methods for second order elliptic problems in two and three dimensions. The methods use piecewise polynomials of degree $k\geqslant 0$ for both the flux and numerical trace, and piecewise polynomials of degree $k+1 $ for the potential. We establish error estimates for the numerical flux and potential under the minimal regularity condition. Moreover, we construct a local postprocessing for the flux, which produces a numerical flux with better conservation. Numerical experiments in two-space dimensions confirm our theoretical results.

Binjie Li, Hao Luo, Xiaoping Xie

This paper analyzes a space-time finite element method for fractional wave problems. The method uses a Petrov-Galerkin type time-stepping scheme to discretize the time fractional derivative of order $ γ$ ($1<γ<2$). We establish the stability of this method, and derive the optimal convergence in the $ H^1(0,T;L^2(Ω)) $-norm and suboptimal convergence in the discrete $ L^\infty(0,T;H_0^1(Ω)) $-norm. Furthermore, we discuss the performance of this method in the case that the solution has singularity at $ t= 0 $, and show that optimal convergence rate with respect to the $ H^1(0,T;L^2(Ω)) $-norm can still be achieved by using graded grids in the time discretization. Finally, numerical experiments are performed to verify the theoretical results.

Chao Chao Yang, Tao Wang, Xiaoping Xie

This paper develops and analyses numerical approximation for linear-quadratic optimal control problem governed by elliptic interface equations. We adopt variational discretization concept to discretize optimal control problem, and apply an interface-unfitted finite element method due to [A. Hansbo and P. Hansbo. An unfitted finite element method, based on Nitsche's method, for elliptic interface problems. Comput. Methods Appl. Mech. Engrg., 191(47-48): 5537-5552, 2002] to discretize corresponding state and adjoint equations, where piecewise cut basis functions around interface are enriched into standard conforming finite element space. Optimal error estimates in both $L^2$ norm and a mesh-dependent norm are derived for optimal state, co-state and control under different regularity assumptions. Numerical results verify the theoretical results.

Tao Wang, Chao Chao Yang, Xiaoping Xie

This paper analyzes two eXtended finite element methods (XFEMs) for linear quadratic optimal control problems governed by Poisson equation in non-convex domains. We follow the variational discretization concept to discretize the continuous problems, and apply an XFEM with a cut-off function and a classic XFEM with a fixed enrichment area to discretize the state and co-state equations. Optimal error estimates are derived for the state, co-state and control. Numerical results confirm our theoretical results.

Xiao Zhang, Xiaoping Xie, Shiquan Zhang

The embedded discontinuous Galerkin (EDG) method by Cockburn et al. [SIAM J. Numer. Anal., 2009, 47(4), 2686-2707] is obtained from the hybridizable discontinuous Galerkin method by changing the space of the Lagrangian multiplier from discontinuous functions to continuous ones, and adopts piecewise polynomials of equal degrees on simplex meshes for all variables. In this paper, we analyze a new EDG method for second order elliptic problems on polygonal/polyhedral meshes. By using piecewise polynomials of degrees $k+1$, $k+1$, $k$ ($k\geq 0$) to approximate the potential, numerical trace and flux, respectively, the new method is shown to yield optimal convergence rates for both the potential and flux approximations. Numerical experiments are provided to confirm the theoretical results.

Tao Wang, Chaochao Yang, Xiaoping Xie

For the optimal control problem governed by elliptic equations with interfaces, we present a numerical method based on the Hansbo's Nitsche-XFEM. We followed the Hinze's variational discretization concept to discretize the continuous problem on a uniform mesh. We derive optimal error estimates of the state, co-state and control both in mesh dependent norm and L2 norm. In addition, our method is suitable for the model with non-homogeneous interface condition. Numerical results confirmed our theoretical results, with the implementation details discussed.

Gang Chen, Xiaoping Xie

This paper proposes and analyzes a class of new weak Galerkin (WG) finite element methods for 2- and 3-dimensional linear elasticity problems. The methods use discontinuous piecewise-polynomial approximations of degrees $k(\geq 0)$ for the stress, $k+1$ for the displacement, and a continuous piecewise-polynomial approximation of degree $k+1$ for the displacement trace on the inter-element boundaries, respectively. After the local elimination of unknowns defined in the interior of elements, the WG methods result in SPD systems where the unknowns are only the degrees of freedom describing the continuous trace approximation. We show that the proposed methods are robust in the sense that the derived a priori error estimates are optimal and uniform with respect to the Lamé constant $λ$. Numerical experiments confirm the theoretical results.

Binjie Li, Hao Luo, Xiaoping Xie

This paper analyzes a time-stepping discontinuous Galerkin method for modified anomalous subdiffusion problems with two time fractional derivatives of orders $ α$ and $ β$ ($ 0 < α< β< 1 $). The stability of this method is established, the temporal accuracy of $ O(τ^{m+1-β/2}) $ is derived, where $m$ denotes the degree of polynomials for the temporal discretization. It is shown that, even the solution has singularity near $ t = {0+} $, this temporal accuracy can still be achieved by using the graded temporal grids. Numerical experiments are performed to verify the theoretical results.

Shaohong Du, Xiaoping Xie

For adaptive mixed finite element methods (AMFEM), we first introduce the data oscillation to analyze, without the restriction that the inverse of the coefficient matrix of the partial differential equations (PDEs) is a piecewise polynomial matrix, efficiency of the a posteriori error estimator Presented by Carstensen [Math. Comput., 1997, 66: 465-476] for Raviart-Thomas, Brezzi-Douglas-Morini, Brezzi-Douglas-Fortin-Marini elements. Second, we prove that the sum of the stress variable error in a weighted norm and the scaled error estimator is of geometric decay, namely, it reduces with a fixed factor between two successive adaptive loops, up to an oscillation of the right-hand side term of the PDEs. Finally, with the help of this geometric decay, we show that the stress variable error in a weighted norm plus the oscillation of data yields a decay rate in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity

Shuaijun Liu, Xiaoping Xie

In this paper, we propose two monolithic fully discrete finite element methods for fluid-structure interaction (FSI) based on a novel Piola-type Arbitrary Lagrangian-Eulerian (ALE) mapping. For the temporal discretization, we apply the backward Euler method to both the non-conservative and conservative formulations. For the spatial discretization, we adopt arbitrary order hybridizable discontinuous Galerkin (HDG) methods for the incompressible Navier-Stokes and linear elasticity equations, and a continuous Galerkin (CG) method for the fluid mesh movement. We derive stability results for both the temporal semi-discretization and the fully discretization, and show that the velocity approximations of the fully discrete schemes are globally divergence-free. Several numerical experiments are performed to verify the performance of the proposed methods.

Qijia Zhai, Pengtao Sun, Xiaoping Xie et al.

In this paper, a meshfree method using physics-informed neural networks (PINNs) is developed for solving two-phase flow problems with moving interfaces, where two immiscible fluids bearing different material properties, are separated by a dynamically evolving interface and interact with each other through interface conditions. Two kinds of distinct scenarios of interface motion are addressed: the prescribed interface motion whose moving velocity is explicitly given, and the solution-driven interface motion whose evolution is determined by the velocity field of two-phase flow. Based upon piecewise deep neural networks and spatiotemporal sampling points/training set in each fluid subdomain, the proposed PINNs framework reformulates the two-phase flow moving interface problem as a least-squares (LS) minimization problem, which involves all residuals of governing equations, interface conditions, boundary conditions and initial conditions. Furthermore, approximation properties of the proposed PINNs approach are analyzed rigorously for the presented two-phase flow model by employing the Reynolds transport theorem in evolving domains, moreover, a comprehensive error estimation is provided to account for additional complexities introduced by the moving interface and the coupling between fluid dynamics and interface evolution. Numerical experiments are carried out to illustrate the effectiveness of the proposed PINNs approach for various configurations of two-phase flow moving interface problems, and to validate the theoretical findings as well. A practical guidance is thus provided for an efficient training set distribution when applying the proposed PINNs approach to two-phase flow moving interface problems in practice.

Maoyuan Xu, Xiaoping Xie

Deep learning based methods have achieved the state-of-the-art performance in image denoising. In this paper, a deep learning based denoising method is proposed and a module called fusion block is introduced in the convolutional neural network. For this so-called Noise Fusion Convolutional Neural Network (NFCNN), there are two branches in its multi-stage architecture. One branch aims to predict the latent clean image, while the other one predicts the residual image. A fusion block is contained between every two stages by taking the predicted clean image and the predicted residual image as a part of inputs, and it outputs a fused result to the next stage. NFCNN has an attractive texture preserving ability because of the fusion block. To train NFCNN, a stage-wise supervised training strategy is adopted to avoid the vanishing gradient and exploding gradient problems. Experimental results show that NFCNN is able to perform competitive denoising results when compared with some state-of-the-art algorithms.

Tao Wang, Chaochao Yang, Xiaoping Xie

This paper analyzes an interface-unfitted numerical method for distributed optimal control problems governed by elliptic interface equations. We follow the variational discretization concept to discretize the optimal control problems, and apply a Nitsche-eXtended finite element method to discretize the corresponding state and adjoint equations, where piecewise cut basis functions around the interface are enriched into the standard linear element space. Optimal error estimates of the state, co-state and control in a mesh-dependent norm and the $L^2$ norm are derived. Numerical results are provided to verify the theoretical results.

Binjie Li, Hao Luo, Xiaoping Xie

This paper develops a high-accuracy algorithm for time fractional wave problems, which employs a spectral method in the temporal discretization and a finite element method in the spatial discretization. Moreover, stability and convergence of this algorithm are derived, and numerical experiments are performed, demonstrating the exponential decay in the temporal discretization error provided the solution is sufficiently smooth.