arXiv subjects

Xiaoping Xie

Publications and source records attributed to Xiaoping Xie.

At least 55 records · Page 3Linked to original sources

arXiv2020

Discontinuous Galerkin method for a distributed optimal control problem governed by a time fractional diffusion equation

This paper is devoted to the numerical analysis of a control constrained distributed optimal control problem subject to a time fractional diffusion equation with non-smooth initial data. The solutions of state and co-state are decomposed into singular and regular parts, and some growth estimates are obtained for the singular parts. Following the variational discretization concept, a full discretization is applied to the corresponding state and co-state equations by using linear conforming finite element method in space and piecewise constant discontinuous Galerkin method in time. By the growth estimates, error estimates are derived with non-smooth initial data. In particular, graded temporal grids are used to obtain the first-order temporal accuracy. Finally, numerical experiments are performed to verify the theoretical results.

math.NA↗

arXiv2020

High order mixed finite elements with mass lumping for elasticity on triangular grids

A family of conforming mixed finite elements with mass lumping on triangular grids are presented for linear elasticity. The stress field is approximated by symmetric $H({\rm div})-P_k (k\geq 3)$ polynomial tensors enriched with higher order bubbles so as to allow mass lumping, which can be viewed as the Hu-Zhang elements enriched with higher order interior bubble functions. The displacement field is approximated by $C^{-1}-P_{k-1}$ polynomial vectors enriched with higher order terms to ensure the stability condition. For both the proposed mixed elements and their mass lumping schemes, optimal error estimates are derived for the stress with $H(\rm div)$ norm and the displacement with $L^2$ norm. Numerical results confirm the theoretical analysis.

math.NA↗

arXiv2020

Numerical analysis of two Galerkin discretizations with graded temporal grids for fractional evolution equations

Two numerical methods with graded temporal grids are analyzed for fractional evolution equations. One is a low-order discontinuous Galerkin (DG) discretization in the case of fractional order $0<α<1$, and the other one is a low-order Petrov Galerkin (PG) discretization in the case of fractional order $1<α<2$. By a new duality technique, pointwise-in-time error estimates of first-order and $ (3-α) $-order temporal accuracies are respectively derived for DG and PG, under reasonable regularity assumptions on the initial value. Numerical experiments are performed to verify the theoretical results.

math.NA↗

arXiv2019

Analysis of the L1 scheme for fractional wave equations with nonsmooth data

This paper analyzes the well-known L1 scheme for fractional wave equations with nonsmooth data. A new stability estimate is obtained, and the temporal accuracy $ \mathcal O(τ^{3-α}) $ is derived for the nonsmooth initial data. In addition, a modified L1 scheme is proposed, and stability and temporal accuracy $ \mathcal O(τ^2) $ are derived for this scheme with nonsmooth initial data. The convergence of the two schemes in the inhomogeneous case is also established. Finally, numerical experiments are performed to verify the theoretical results.

math.NA↗

arXiv2019

Extended HDG methods for second order elliptic interface problems

In this paper, we propose two arbitrary order eXtended hybridizable Discontinuous Galerkin (X-HDG) methods for second order elliptic interface problems in two and three dimensions. The first X-HDG method applies to any piecewise $C^2$ smooth interface. It uses piecewise polynomials of degrees $k$ $(k>= 1)$ and $k-1$ respectively for the potential and flux approximations in the interior of elements inside the subdomains, and piecewise polynomials of degree $ k$ for the numerical traces of potential on the inter-element boundaries inside the subdomains. Double value numerical traces on the parts of interface inside elements are adopted to deal with the jump condition. The second X-HDG method is a modified version of the first one and applies to any fold line/plane interface, which uses piecewise polynomials of degree $ k-1$ for the numerical traces of potential. The X-HDG methods are of the local elimination property, then lead to reduced systems which only involve the unknowns of numerical traces of potential on the inter-element boundaries and the interface. Optimal error estimates are derived for the flux approximation in $L^2$ norm and for the potential approximation in piecewise $H^1$ seminorm without requiring "sufficiently large" stabilization parameters in the schemes. In addition, error estimation for the potential approximation in $L^2$ norm is performed using dual arguments. Finally, we provide several numerical examples to verify the theoretical results.

math.NA↗

arXiv2019

Numerical analysis of a semilinear fractional diffusion equation

This paper considers the numerical analysis of a semilinear fractional diffusion equation with nonsmooth initial data. A new Grönwall's inequality and its discrete version are proposed. By the two inequalities, error estimates in three Sobolev norms are derived for a spatial semi-discretization and a full discretization, which are optimal with respect to the regularity of the solution. A sharp temporal error estimate on graded temporal grids is also rigorously established. In addition, the spatial accuracy $\scriptstyle O(h^2(t^{-α} + \ln(1/h)\!)\!) $ in the pointwise $ \scriptstyle L^2(Ω) $-norm is obtained for a spatial semi-discretization. Finally, several numerical results are provided to verify the theoretical results.

math.NA↗

arXiv2019

Analysis of a time-stepping discontinuous Galerkin method for fractional diffusion-wave equation with nonsmooth data

This paper analyzes a time-stepping discontinuous Galerkin method for fractional diffusion-wave problems. This method uses piecewise constant functions in the temporal discretization and continuous piecewise linear functions in the spatial discretization. Nearly optimal convergence rate with respect to the regularity of the solution is established when the source term is nonsmooth, and nearly optimal convergence rate $ \ln(1/τ)(\sqrt{\ln(1/h)} h^2+τ)$ is derived under appropriate regularity assumption on the source term. Convergence is also established without smoothness assumption on the initial value. Finally, numerical experiments are performed to verify the theoretical results.

math.NA↗

arXiv2019

Convergence analysis of a Petrov-Galerkin method for fractional wave problems with nonsmooth data

This paper analyzes the convergence of a Petrov-Galerkin method for time fractional wave problems with nonsmooth data. Well-posedness and regularity of the weak solution to the time fractional wave problem are firstly established. Then an optimal convergence analysis with nonsmooth data is derived. Moreover, several numerical experiments are presented to validate the theoretical results.

math.NA↗

arXiv2018

A Nitsche-eXtended finite element method for distributed optimal control problems of elliptic interface equations

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.

math.NA↗

arXiv2018

Extended finite element methods for optimal control problems governed by Poisson equation in non-convex domains

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.

math.NA↗

arXiv2018

An interface-unfitted finite element method for elliptic interface optimal control problem

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.

math.NA↗

arXiv2018

Nitsche-XFEM for optimal control problems governed by elliptic PDEs with interfaces

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.

math.NA↗

arXiv2018

Self-ordering induces multiple topological transitions for elastic waves in phononic crystals

Topological defects with symmetry-breaking phase transitions have captured much attention. Vortex generated by topological defects exhibits exotic properties and its flow direction can be switched by altering the spin configurations. Contrary to electromagnetic and acoustic domains, the topological transport of elastic waves in periodic structures with topological defects is not well explored due to the mode conversion between the longitudinal and transverse modes. Here, we propose an elastic topological insulator with spontaneously broken symmetry based on the topological theory of defects and homotopy theory. Multiple topological transitions for elastic waves are achieved by topologically modifying the ellipse orientation in a triangular lattice of elliptical cylinders. The solid system, independent of the number of molecules in order parameter space, breaks through the limit of the point-group symmetry to emulate elastic pseudospin-orbit coupling. The transport robustness of the edge states is experimentally demonstrated. Our approach provides new possibilities for controlling and transporting elastic waves.

physics.class-ph↗

arXiv2018

Analysis of a time-stepping scheme for time fractional diffusion problems with nonsmooth data

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.

math.NA↗

arXiv2018

A space-time finite element method for fractional wave problems

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.

math.NA↗

arXiv2017

Analysis of a time-stepping discontinuous Galerkin method for modified anomalous subdiffusion problems

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.

math.NA↗

arXiv2017

An Optimal Embedded Discontinuous Galerkin Method for Second-Order Elliptic Problems

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.

math.NA↗

arXiv2017

Robust weak Galerkin finite element methods for linear elasticity with continuous displacement trace approximation

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.

math.NA↗