Xu Zhang

Xu Zhang
Oklahoma State University - Stillwater | Oklahoma State · Department of Mathematics

PhD

About

41
Publications
4,287
Reads
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800
Citations
Citations since 2017
24 Research Items
692 Citations
2017201820192020202120222023020406080100120140
2017201820192020202120222023020406080100120140
2017201820192020202120222023020406080100120140
2017201820192020202120222023020406080100120140
Introduction
Xu Zhang currently works at the Department of Mathematics, Oklahoma State University. Xu does research in Applied Mathematics. His current project is numerical methods for interface problems.
Additional affiliations
August 2019 - present
Oklahoma State University - Stillwater
Position
  • Professor (Assistant)
August 2016 - August 2019
Mississippi State University
Position
  • Professor (Assistant)
August 2013 - May 2016
Purdue University
Position
  • Golomb Visiting Assistant Professor

Publications

Publications (41)
Article
This paper designs and analyzes a new and stable Petrov-Galerkin (PG) immersed finite element method (IFEM) for the second-order elliptic interface problems by introducing stabilization terms based on the classical PG-IFEM, which lacks the local positivity. The Petrov-Galerkin immersed finite element method uses the immersed finite element function...
Article
Weak scaling performance of a recently developed fully kinetic, 3-D parallel immersed-finite-element particle-in-cell framework, namely PIFE-PIC, was investigated. A nominal 1-D plasma charging problem, the lunar photoelectron sheath at a low Sun elevation angle, was chosen to validate PIFE-PIC against recently derived semi-analytic solutions of a...
Article
Immersed finite element methods are designed to solve interface problems on interface-unfitted meshes. However, most of the study, especially analysis, is mainly limited to the two-dimension case. In this paper, we provide an a priori analysis for the trilinear immersed finite element method to solve three-dimensional elliptic interface problems on...
Article
In this paper, we introduce a class of lowest-order nonconforming immersed finite element (IFE) methods for solving two-dimensional Stokes interface problems. The proposed methods do not require the solution mesh to align with the fluid interface and can use either triangular or rectangular meshes. On triangular meshes, the Crouzeix–Raviart element...
Article
Full-text available
In this article, we develop a Taylor-Hood immersed finite element (IFE) method to solve two-dimensional Stokes interface problems. The P2-P1 local IFE spaces are constructed using the least-squares approximation on an enlarged fictitious element. The partially penalized IFE method with ghost penalty is employed for solving Stoke interface problems....
Article
In this article, we develop a new mixed immersed finite element discretization for two-dimensional unsteady Stokes interface problems with unfitted meshes. The proposed IFE spaces use conforming linear elements for one velocity component and non-conforming linear elements for the other velocity component. The pressure is approximated by piecewise c...
Preprint
This paper presents a recently developed particle simulation code package PIFE-PIC, which is a novel three-dimensional (3-D) Parallel Immersed-Finite-Element (IFE) Particle-in-Cell (PIC) simulation model for particle simulations of plasma-material interactions. This framework is based on the recently developed non-homogeneous electrostatic IFE-PIC...
Preprint
Full-text available
In this paper, we develop and analyze a trilinear immersed finite element method for solving three-dimensional elliptic interface problems. The proposed method can be utilized on interface-unfitted meshes such as Cartesian grids consisting of cuboids. We establish the trace and inverse inequalities for trilinear IFE functions for interface elements...
Preprint
In this article, we introduce a new partially penalized immersed finite element method (IFEM) for solving elliptic interface problems with multi-domains and triple-junction points. We construct new IFE functions on elements intersected with multiple interfaces or with triple-junction points to accommodate interface jump conditions. For non-homogene...
Article
Full-text available
In this article, we introduce a new partially penalized immersed finite element method (IFEM) for solving elliptic interface problems with multi-domain and triple-junction points. We construct new IFE functions on elements intersected with multiple interfaces or with triple-junction points to accommodate interface jump conditions. For non-homogeneo...
Article
Full-text available
In this paper we introduce and analyze the residual-based a posteriori error estimation of the partially penalized immersed finite element method for solving elliptic interface problems. The immersed finite element method can be naturally utilized on interface-unfitted meshes. Our a posteriori error estimate is proved to be both reliable and effici...
Preprint
In this paper we introduce and analyze the residual-based a posteriori error estimation of the partially penalized immersed finite element method for solving elliptic interface problems. The immersed finite element method can be naturally utilized on interface-unfitted meshes. Our a posteriori error estimate is proved to be both reliable and effici...
Article
Full-text available
We present a high order immersed finite element (IFE) method for solving 1D parabolic interface problems. These methods allow the solution mesh to be independent of the interface. Time marching schemes including Backward-Eulerand Crank-Nicolson methods are implemented to fully discretize the system. Numerical examples are provided to test the perfo...
Article
Full-text available
Interface problems have wide applications in modern scientific research. Obtaining accurate numerical solutions of multi-domain problems involving triple junction conditions remains a significant challenge. In this paper, we develop an efficient finite element method based on non-body-fitting meshes for solving multi-domain elliptic interface probl...
Article
Full-text available
New immersed finite element (IFE) methods are developed for second-order elliptic problems with discontinuous diffusion coefficient. IFE spaces are constructed based on the rotated--$Q_1$ nonconforming finite elements with the edge midpoint value and the edge mean value degrees of freedom. Approximation capability of these IFE spaces is analyzed by...
Article
Full-text available
In this paper, we develop an efficient numerical scheme for solving one-dimensional hyperbolic interface problems. The immersed finite element (IFE) method is used for spatial discretization, which allows the solution mesh to be independent of the interface. Consequently, a fixed uniform mesh can be used throughout the entire simulation. The method...
Article
In this paper, we present an immersed weak Galerkin method for solving second-order elliptic interface problems. The proposed method does not require the meshes to be aligned with the interface. Consequently, uniform Cartesian meshes can be used for nontrivial interfacial geometry. We show the existence and uniqueness of the numerical algorithm, an...
Article
Full-text available
The original version of this article unfortunately contained an error. © 2017 Springer Science+Business Media, LLC, part of Springer Nature
Article
Full-text available
In this paper, we introduce a class of high order immersed finite volume methods (IFVM) for one-dimensional interface problems. We show the optimal convergence of IFVM in \(H^1\)- and \(L^2\)-norms. We also prove some superconvergence results of IFVM. To be more precise, the IFVM solution is superconvergent of order \(p+2\) at the roots of generali...
Article
Full-text available
In this article, we study superconvergence properties of immersed finite element methods for the one dimensional elliptic interface problem. Due to low global regularity of the solution, classical superconvergence phenomenon for finite element methods disappears unless the discontinuity of the coefficient is resolved by partition. We show that imme...
Article
Full-text available
The particle-in-cell (PIC) method has been widely used for plasma simulation, because of its noise-reduction capability and moderate computational cost. The immersed finite element (IFE) method is efficient for solving interface problems on Cartesian meshes, which is desirable for PIC method. The combination of these two methods provides an effecti...
Article
Full-text available
In this paper, we use a unified framework introduced in [3] to study two classes of nonconforming immersed finite element (IFE) spaces with integral value degrees of freedom. The shape functions on interface elements are piecewise polynomials defined on sub-elements separated either by the actual interface or its line approximation. In this unified...
Article
In this paper, we analyze convergence and supercloseness properties of a class of weak Galerkin (WG) finite element methods for solving second-order elliptic problems. It is shown that the WG solution is superclose to the Lagrange type interpolation using Lobatto points. This supercloseness behavior is obtained through some newly designed stabiliza...
Article
Interface problems arise in many physical and engineering simulations involving multiple materials. Periodic structures often appear in simulations with large or even unbounded domain, such as magnetostatic/electrostatic field simulations. Immersed finite element (IFE) methods are efficient tools to solve interface problems on a Cartesian mesh, whi...
Article
In this article, interior penalty discontinuous Galerkin methods using immersed finite element functions are employed to solve parabolic interface problems. Typical semi-discrete and fully discrete schemes are presented and analyzed. Optimal convergence for both semi-discrete and fully discrete schemes are proved. Some numerical experiments are pro...
Article
Full-text available
In this paper, we derive a priori error estimates for a class of interior penalty discontinuous Galerkin (DG) methods using immersed finite element (IFE) functions for a classic second-order elliptic interface problem. The error estimation shows that these methods can converge optimally in a mesh-dependent energy norm. The combination of IFEs and D...
Article
This article presents new immersed finite element (IFE) methods for solving the popular second order elliptic interface problems on structured Cartesian meshes even if the involved interfaces have nontrivial geometries. These IFE methods contain extra stabilization terms introduced only at interface edges for penalizing the discontinuity in IFE fun...
Article
Full-text available
This article is to discuss the bilinear and linear immersed finite element (IFE) solutions generated fromthe algebraicmultigrid solver for both stationary andmoving interface problems. For the numerical methods based on finite difference formulation and a structured mesh independent of the interface, the stiffness matrix of the linear system is usu...
Article
Full-text available
This article extends the finite element method of lines to a parabolic initial boundary value problem whose diffusion coefficient is discontinuous across an interface that changes with respect to time. The method presented here uses immersed finite element (IFE) functions for the discretization in spatial variables that can be carried out over a fi...
Article
Full-text available
This article presents an immersed finite element method of lines for solving parabolic moving interface problems with a non-homogeneous flux jump. The immersed finite elements are used for spatial discretization, which allow the material interface to be embedded in the interior of elements in the mesh. This feature makes it possible to employ the m...
Article
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This article presents three Crank‐Nicolson‐type immersed finite element (IFE) methods for solving parabolic equations whose diffusion coefficient is discontinuous across a time dependent interface. These methods can use a fixed mesh because IFEs can handle interface jump conditions without requiring the mesh to be aligned with the interface. These...
Article
This article is to discuss the linear (which was proposed in and ) and bilinear immersed finite element (IFE) methods for solving planar elasticity interface problems with structured Cartesian meshes. Basic features of linear and bilinear IFE functions, including the unisolvent property, will be discussed. While both methods have comparable accurac...
Article
Full-text available
In order to solve a non-stationary Stokes–Darcy model with Beavers–Joseph interface condition, two non-iterative domain decomposition methods are proposed. At each time step, results from previous time steps are utilized to approximate the information on the interface and decouple the two physics. Both of the two methods are parallel. Numerical res...
Article
Based on a weighted average of the modified Hellinger–Reissner principle and its dual, the combined hybrid finite element (CHFE) method was originally proposed with a combination parameter limited in the interval (0,1). In actual computation this parameter plays an important role in adjusting the energy error of discretization models. In this paper...
Article
Full-text available
The authors consider fully discrete finite element methods for the fluid-structure interaction system. For the time discretization, a backward difference algorithm and composite left rectangular methods are adopted to approximate the continuous derivative and for integration with respect to t, respectively. Existence and uniqueness of finite elemen...

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