arXiv subjects

Jinchao Xu

Publications and source records attributed to Jinchao Xu.

At least 91 records · Page 5Linked to original sources

arXiv2018

A Unified Study of Continuous and Discontinuous Galerkin Methods

A unified study is presented in this paper for the design and analysis of different finite element methods (FEMs), including conforming and nonconforming FEMs, mixed FEMs, hybrid FEMs,discontinuous Galerkin (DG) methods, hybrid discontinuous Galerkin (HDG) methods and weak Galerkin (WG) methods. Both HDG and WG are shown to admit inf-sup conditions that hold uniformly with respect to both mesh and penalization parameters. In addition, by taking the limit of the stabilization parameters, a WG method is shown to converge to a mixed method whereas an HDG method is shown to converge to a primal method. Furthermore, a special class of DG methods, known as the mixed DG methods, is presented to fill a gap revealed in the unified framework.

math.NA↗

arXiv2018

Nonconforming Finite Element Spaces for $2m$-th Order Partial Differential Equations on $\mathbb{R}^n$ Simplicial Grids When $m=n+1$

In this paper, we propose a family of nonconforming finite elements for $2m$-th order partial differential equations in $\mathbb{R}^n$ on simplicial grids when $m=n+1$. This family of nonconforming elements naturally extends the elements proposed by Wang and Xu [Math. Comp. 82(2013), pp. 25-43] , where $m \leq n$ is required. We prove the unisolvent property by induction on the dimensions using the similarity properties of both shape function spaces and degrees of freedom. The proposed elements have approximability, pass the generalized patch test and hence converge. We also establish quasi-optimal error estimates in the broken $H^3$ norm for the 2D nonconforming element. In addition, we propose an $H^3$ nonconforming finite element that is robust for the sixth order singularly perturbed problems in 2D. These theoretical results are further validated by the numerical tests for the 2D tri-harmonic problem.

math.NA↗

arXiv2017

On the Stability and Accuracy of Partially and Fully Implicit Schemes for Phase Field Modeling

We study in this paper the accuracy and stability of partially and fully implicit schemes for phase field modeling. Through theoretical and numerical analysis of Allen-Cahn and Cahn-Hillard models, we investigate the potential problems of using partially implicit schemes, demonstrate the importance of using fully implicit schemes and discuss the limitation of energy stability that are often used to evaluate the quality of a numerical scheme for phase-field modeling. In particular, we make the following observations: 1. a convex splitting scheme (CSS in short) can be equivalent to some fully implicit scheme (FIS in short) with a much different time scaling and thus it may lack numerical accuracy; 2. most implicit schemes (in discussions) are energy-stable if the time-step size is sufficiently small; 3. a traditionally known conditionally energy-stable scheme still possess an unconditionally energy-stable physical solution; 4. an unconditionally energy-stable scheme is not necessarily better than a conditionally energy-stable scheme when the time step size is not small enough; 5. a first-order FIS for the Allen-Cahn model can be devised so that the maximum principle will be valid on the discrete level and hence the discrete phase variable satisfies $|u_h(x)|\le 1$ for all $x$ and, furthermore, the linearized discretized system can be effectively preconditioned by discrete Poisson operators.

math.NA↗

arXiv2017

Structure-preserving Finite Element Methods for Stationary MHD Models

In this paper, we develop a class of mixed finite element scheme for stationary magnetohydrodynamics (MHD) models, using magnetic field $\bm B$ and current density $\bm j$ as the discretization variables. We show that the Gauss's law for the magnetic field, namely $\nabla\cdot\bm{B}=0$, and the energy law for the entire system are exactly preserved in the finite element schemes. Based on some new basic estimates for $H^{h}(\mathrm{div})$, we show that the new finite element scheme is well-posed. Furthermore, we show the existence of solutions to the nonlinear problems and the convergence of Picard iterations and finite element methods under some conditions.

math.NA↗

arXiv2017

A Derandomized Algorithm for RP-ADMM with Symmetric Gauss-Seidel Method

For multi-block alternating direction method of multipliers(ADMM), where the objective function can be decomposed into multiple block components, we show that with block symmetric Gauss-Seidel iteration, the algorithm will converge quickly. The method will apply a block symmetric Gauss-Seidel iteration in the primal update and a linear correction that can be derived in view of Richard iteration. We also establish the linear convergence rate for linear systems.

math.OC↗

arXiv2017

A unified approach to the design and analysis of AMG

In this work, we present a general framework for the design and analysis of two-level AMG methods. The approach is to find a basis for locally optimal or quasi-optimal coarse space, such as the space of constant vectors for standard discretizations of scalar elliptic partial differential equations. The locally defined basis elements are glued together using carefully designed linear extension maps to form a global coarse space. Such coarse spaces, constructed locally, satisfy global approximation property and by estimating the local Poincar{\' e} constants, we obtain sharp bounds on the convergence rate of the resulting two-level methods. To illustrate the use of the theoretical framework in practice, we prove the uniform convergence of the classical two level AMG method for finite element discretization of a jump coefficient problem on a shape regular mesh.

math.NA↗

arXiv2017

Interior Penalty Mixed Finite Element Methods of Any Order in Any Dimension for Linear Elasticity with Strongly Symmetric Stress Tensor

We propose two classes of mixed finite elements for linear elasticity of any order, with interior penalty for nonconforming symmetric stress approximation. One key point of our method is to introduce some appropriate nonconforming face-bubble spaces based on the local decomposition of discrete symmetric tensors, with which the stability can be easily established. We prove the optimal error estimate for both displacement and stress by adding an interior penalty term. The elements are easy to be implemented thanks to the explicit formulations of its basis functions. Moreover, the methods can be applied to arbitrary simplicial grids for any spatial dimension in a unified fashion. Numerical tests for both 2D and 3D are provided to validate our theoretical results.

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arXiv2017

New Hybridized Mixed Methods for Linear Elasticity and Optimal Multilevel Solvers

In this paper, we present a family of new mixed finite element methods for linear elasticity for both spatial dimensions $n=2,3$, which yields a conforming and strongly symmetric approximation for stress. Applying $\mathcal{P}_{k+1}-\mathcal{P}_k$ as the local approximation for the stress and displacement, the mixed methods achieve the optimal order of convergence for both the stress and displacement when $k \ge n$. For the lower order case $(n-2\le k<n)$, the stability and convergence still hold on some special grids. The proposed mixed methods are efficiently implemented by hybridization, which imposes the inter-element normal continuity of the stress by a Lagrange multiplier. Then, we develop and analyze multilevel solvers for the Schur complement of the hybridized system in the two dimensional case. Provided that no nearly singular vertex on the grids, the proposed solvers are proved to be uniformly convergent with respect to both the grid size and Poisson's ratio. Numerical experiments are provided to validate our theoretical results.

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arXiv2017

Multiphase Allen-Cahn and Cahn-Hilliard Models and Their Discretizations with the Effect of Pairwise Surface Tensions

In this paper, the mathematical properties and numerical discretizations of multiphase models that simulate the phase separation of an $N$-component mixture are studied. For the general choice of phase variables, the unisolvent property of the coefficient matrix involved in the $N$-phase models based on the pairwise surface tensions is established. Moreover, the symmetric positive-definite property of the coefficient matrix on an $(N-1)$-dimensional hyperplane --- which is of fundamental importance to the well-posedness of the models --- can be proved equivalent to some physical condition for pairwise surface tensions. The $N$-phase Allen-Cahn and $N$-phase Cahn-Hilliard equations can then be derived from the free-energy functional. A natural property is that the resulting dynamics of concentrations are independent of phase variables chosen. Finite element discretizations for $N$-phase models can be obtained as a natural extension of the existing discretizations for the two-phase model. The discrete energy law of the numerical schemes can be proved and numerically observed under some restrictions pertaining to time step size. Numerical experiments including the spinodal decomposition and the evolution of triple junctions are described in order to investigate the effect of pairwise surface tensions.

math-ph↗

arXiv2016

Algebraic Multigrid Methods

This paper is to give an overview of AMG methods for solving large scale systems of equations such as those from the discretization of partial differential equations. AMG is often understood as the acronym of "Algebraic Multi-Grid", but it can also be understood as "Abstract Muti-Grid". Indeed, as it demonstrates in this paper, how and why an algebraic multigrid method can be better understood in a more abstract level. In the literature, there are a variety of different algebraic multigrid methods that have been developed from different perspectives. In this paper, we try to develop a unified framework and theory that can be used to derive and analyze different algebraic multigrid methods in a coherent manner. Given a smoother $R$ for a matrix $A$, such as Gauss-Seidel or Jacobi, we prove that the optimal coarse space of dimension $n_c$ is the span of the eigen-vectors corresponding to the first $n_c$ eigenvalues of $\bar RA$ (with $\bar R=R+R^T-R^TAR$). We also prove that this optimal coarse space can be obtained by a constrained trace-minimization problem for a matrix associated with $\bar RA$ and demonstrate that coarse spaces of most of existing AMG methods can be viewed some approximate solution of this trace-minimization problem. Furthermore, we provide a general approach to the construction of a quasi-optimal coarse space and we prove that under appropriate assumptions the resulting two-level AMG method for the underlying linear system converges uniformly with respect to the size of the problem, the coefficient variation, and the anisotropy. Our theory applies to most existing multigrid methods, including the standard geometric multigrid method, the classic AMG, energy-minimization AMG, unsmoothed and smoothed aggregation AMG, and spectral AMGe.

math.NA↗

arXiv2016

Error estimates for structure-preserving discretization of the incompressible MHD system

In this paper, we carry out the error analysis for the structure-preserving discretization of the incompressible MHD system. This system, as a coupled system of Navier-Stokes equations and Maxwell's equations, is nonlinear. We use its energy estimate and the underlying physical structure to facilitate the error analysis. Under certain CFL conditions, we prove the optimal order of convergence. To support the theoretical results, we also present numerical tests.

math.NA↗

arXiv2016

High-Order Extended Finite Element Methods for Solving Interface Problems

In this paper, we study arbitrary order extended finite element (XFE) methods based on two discontinuous Galerkin (DG) schemes in order to solve elliptic interface problems in two and three dimensions. Optimal error estimates in the piecewise $H^1$-norm and in the $L^2$-norm are rigorously proved for both schemes. In particular, we have devised a new parameter-friendly DG-XFEM method, which means that no "sufficiently large" parameters are needed to ensure the optimal convergence of the scheme. To prove the stability of bilinear forms, we derive non-standard trace and inverse inequalities for high-order polynomials on curved sub-elements divided by the interface. All the estimates are independent of the location of the interface relative to the meshes. Numerical examples are given to support the theoretical results.

math.NA↗

arXiv2016

Fast multilevel solvers for a class of discrete fourth order parabolic problems

In this paper, we study fast iterative solvers for the solution of fourth order parabolic equations discretized by mixed finite element methods. We propose to use consistent mass matrix in the discretization and use lumped mass matrix to construct efficient preconditioners. We provide eigenvalue analysis for the preconditioned system and estimate the convergence rate of the preconditioned GMRes method. Furthermore, we show that these preconditioners only need to be solved inexactly by optimal multigrid algorithms. Our numerical examples indicate that the proposed preconditioners are very efficient and robust with respect to both discretization parameters and diffusion coefficients. We also investigate the performance of multigrid algorithms with either collective smoothers or distributive smoothers when solving the preconditioner systems.

math.NA↗

arXiv2015

Modeling and Simulation for Fluid-Rotating Structure Interaction

In this paper, we study a dynamic fluid-structure interaction (FSI) model for an elastic structure that is immersed and spinning in the fluid. We develop a linear constitutive model to describe the motion of a rotational elastic structure which is suitable for the application of arbitrary Lagrangian-Eulerian (ALE) method in FSI simulation. Additionally, a novel ALE mapping method is designed to generate the moving fluid mesh while the deformable structure spins in a non-axisymmetric fluid channel. The structure velocity is adopted as the principle unknown to form a monolithic saddle-point system together with fluid velocity and pressure. We discretize the nonlinear saddle-point system with mixed finite element method and Newton's linearization, and prove that the derived saddle-point problem is well-posed. The developed methodology is applied to a self-defined elastic structure and a realistic hydro-turbine under a prescribed angular velocity. Both illustrate the satisfactory numerical results of an elastic structure that is deforming and rotating while interacting with the fluid. The numerical validation is also conducted to demonstrate the modeling consistency.

math.NA↗

arXiv2015

Energetically stable discretizations for charge carrier transport and electrokinetic models

A finite element discretization using a method of lines approached is proposed for approximately solving the Poisson-Nernst-Planck (PNP) equations. This discretization scheme enforces positivity of the computed solutions, corresponding to particle density functions, and a discrete energy estimate is established that resembles the familiar energy law for the PNP system. This energy estimate is extended to finite element solutions to an electrokinetic model, which couples the PNP system with the Navier-Stokes equations. Numerical experiments are conducted to validate convergence of the computed solution and verify the discrete energy estimate.

math.NA↗

arXiv2015

Robust Preconditioners for Incompressible MHD Models

In this paper, we develop two classes of robust preconditioners for the structure-preserving discretization of the incompressible magnetohydrodynamics (MHD) system. By studying the well-posedness of the discrete system, we design block preconditioners for them and carry out rigorous analysis on their performance. We prove that such preconditioners are robust with respect to most physical and discretization parameters. In our proof, we improve the existing estimates of the block triangular preconditioners for saddle point problems by removing the scaling parameters, which are usually difficult to choose in practice. This new technique is not only applicable to the MHD system, but also to other problems. Moreover, we prove that Krylov iterative methods with our preconditioners preserve the divergence-free condition exactly, which complements the structure-preserving discretization. Another feature is that we can directly generalize this technique to other discretizations of the MHD system. We also present preliminary numerical results to support the theoretical results and demonstrate the robustness of the proposed preconditioners.

math.NA↗

arXiv2014

A Cascadic Multigrid Algorithm for Computing the Fiedler Vector of Graph Laplacians

In this paper, we develop a cascadic multigrid algorithm for fast computation of the Fiedler vector of a graph Laplacian, namely, the eigenvector corresponding to the second smallest eigenvalue. This vector has been found to have applications in fields such as graph partitioning and graph drawing. The algorithm is a purely algebraic approach based on a heavy edge coarsening scheme and pointwise smoothing for refinement. To gain theoretical insight, we also consider the related cascadic multigrid method in the geometric setting for elliptic eigenvalue problems and show its uniform convergence under certain assumptions. Numerical tests are presented for computing the Fiedler vector of several practical graphs, and numerical results show the efficiency and optimality of our proposed cascadic multigrid algorithm.

math.NA↗

arXiv2014

Multilevel Preconditioners for Reaction-Diffusion Problems with Discontinuous Coefficients

In this paper, we extend some of the multilevel convergence results obtained by Xu and Zhu in [Xu and Zhu, M3AS 2008], to the case of second order linear reaction-diffusion equations. Specifically, we consider the multilevel preconditioners for solving the linear systems arising from the linear finite element approximation of the problem, where both diffusion and reaction coefficients are piecewise-constant functions. We discuss in detail the influence of both the discontinuous reaction and diffusion coefficients to the performance of the classical BPX and multigrid V-cycle preconditioners.

math.NA↗