arXiv subjects

Xiaobing Feng

Publications and source records attributed to Xiaobing Feng.

86 records · Page 5Linked to original sources

arXiv2012

Local discontinuous Galerkin methods for one-dimensional second order fully nonlinear elliptic and parabolic equations

This paper is concerned with developing accurate and efficient discontinuous Galerkin methods for fully nonlinear second order elliptic and parabolic partial differential equations (PDEs) in the case of one spatial dimension. The primary goal of the paper to develop a general framework for constructing high order local discontinuous Galerkin (LDG) methods for approximating viscosity solutions of these fully nonlinear PDEs which are merely continuous functions by definition. In order to capture discontinuities of the first order derivative $u_x$ of the solution $u$, two independent functions $q_1$ and $q_2$ are introduced to approximate one-sided derivatives of $u$. Similarly, to capture the discontinuities of the second order derivative $u_{xx}$, four independent functions $p_{1}$, $p_{2}$, $p_{3}$, and $p_{4}$ are used to approximate one-sided derivatives of $q_1$ and $q_2$. The proposed LDG framework, which is based on a nonstandard mixed formulation of the underlying PDE, embeds a given fully nonlinear problem into a mostly linear system of equations where the given nonlinear differential operator must be replaced by a numerical operator which allows multiple value inputs of the first and second order derivatives $u_x$ and $u_{xx}$. An easy to verify criterion for constructing "good" numerical operators is also proposed. It consists of a consistency and a generalized monotonicity. To ensure such a generalized monotonicity, the crux of the construction is to introduce the numerical moment in the numerical operator. The proposed framework extends a companion finite difference framework developed by the authors in [9] and allows for the approximation of fully nonlinear PDEs using high order polynomials and non-uniform meshes.

math.NA↗

arXiv2011

The Vanishing Moment Method for Fully Nonlinear Second Order Partial Differential Equations: Formulation, Theory, and Numerical Analysis

The vanishing moment method was introduced by the authors in [37] as a reliable methodology for computing viscosity solutions of fully nonlinear second order partial differential equations (PDEs), in particular, using Galerkin-type numerical methods such as finite element methods, spectral methods, and discontinuous Galerkin methods, a task which has not been practicable in the past. The crux of the vanishing moment method is the simple idea of approximating a fully nonlinear second order PDE by a family (parametrized by a small parameter $\vepsi$) of quasilinear higher order (in particular, fourth order) PDEs. The primary objectives of this book are to present a detailed convergent analysis for the method in the radial symmetric case and to carry out a comprehensive finite element numerical analysis for the vanishing moment equations (i.e., the regularized fourth order PDEs). Abstract methodological and convergence analysis frameworks of conforming finite element methods and mixed finite element methods are first developed for fully nonlinear second order PDEs in general settings. The abstract frameworks are then applied to three prototypical nonlinear equations, namely, the Monge-Ampère equation, the equation of prescribed Gauss curvature, and the infinity-Laplacian equation. Numerical experiments are also presented for each problem to validate the theoretical error estimate results and to gauge the efficiency of the proposed numerical methods and the vanishing moment methodology.

math.NA↗

arXiv2010

Absolutely stable local discontinuous Galerkin methods for the Helmholtz equation with large wave number

Two local discontinuous Galerkin (LDG) methods using some non-standard numerical fluxes are developed for the Helmholtz equation with the first order absorbing boundary condition in the high frequency regime. It is shown that the proposed LDG methods are absolutely stable (hence well-posed) with respect to both the wave number and the mesh size. Optimal order (with respect to the mesh size) error estimates are proved for all wave numbers in the preasymptotic regime. To analyze the proposed LDG methods, they are recasted and treated as (non-conforming) mixed finite element methods. The crux of the analysis is to establish a generalized {\em inf-sup} condition, which holds without any mesh constraint, for each LDG method. The generalized {\em inf-sup} conditions then easily infer the desired absolute stability of the proposed LDG methods. In return, the stability results not only guarantee the well-posedness of the LDG methods but also play a crucial role in the derivation of the error estimates. Numerical experiments, which confirm the theoretical results and compare the proposed two LDG methods, are also presented in the paper.

math.NA↗

arXiv2009

$hp$-discontinuous Galerkin Methods for the Helmholtz Equation with Large Wave Number

This paper develops some interior penalty $hp$-discontinuous Galerkin ($hp$-DG) methods for the Helmholtz equation in two and three dimensions. The proposed $hp$-DG methods are defined using a sesquilinear form which is not only mesh-dependent but also degree-dependent. In addition, the sesquilinear form contains penalty terms which not only penalize the jumps of the function values across the element edges but also the jumps of the first order tangential derivatives as well as jumps of all normal derivatives up to order $p$. Furthermore, to ensure the stability, the penalty parameters are taken as complex numbers with positive imaginary parts. It is proved that the proposed $hp$-discontinuous Galerkin methods are absolutely stable (hence, well-posed). For each fixed wave number $k$, sub-optimal order error estimates in the broken $H^1$-norm and the $L^2$-norm are derived without any mesh constraint. The error estimates and the stability estimates are improved to optimal order under the mesh condition $k^3h^2p^{-1}\le C_0$ by utilizing these stability and error estimates and using a stability-error iterative procedure To overcome the difficulty caused by strong indefiniteness of the Helmholtz problems in the stability analysis for numerical solutions, our main ideas for stability analysis are to make use of a local version of the Rellich identity (for the Laplacian) and to mimic the stability analysis for the PDE solutions given in \cite{cummings00,Cummings_Feng06,hetmaniuk07}, which enable us to derive stability estimates and error bounds with explicit dependence on the mesh size $h$, the polynomial degree $p$, the wave number $k$, as well as all the penalty parameters for the numerical solutions.

math.NA↗

arXiv2009

Analysis of Fully Discrete Finite Element Methods for a System of Differential Equations Modeling Swelling Dynamics of Polymer Gels

The goal of this paper is to develop and analyze some fully discrete finite element methods for a displacement-pressure model modeling swelling dynamics of polymer gels under mechanical constraints. In the model, the swelling dynamics is governed by the solvent permeation and the elastic interaction; the permeation is described by a pressure equation for the solvent, and the elastic interaction is described by displacement equations for the solid network of the gel. By introducing an "elastic pressure" we first present a reformulation of the original model, and then propose a time-stepping scheme which decouples the PDE system at each time step into two sub-problems, one of which is a generalized Stokes problem for the displacement vector field and another is a diffusion problem for a "pseudo-pressure" field. To make such a multiphysical approach feasible, it is vital to discover admissible constraints to resolve the uniqueness issue for both sub-problems. The main advantage of the proposed approach is that it allows one to utilize any convergent Stokes solver together with any convergent diffusion equation solver to solve the polymer gel model. In the paper, the Taylor-Hood mixed finite element method combined with the continuous linear finite element method are used as an example to present the ideas and to demonstrate the viability of the proposed multiphysical approach. It is proved that, under a mesh constraint, both the proposed semi-discrete (in space) and fully discrete methods enjoy some discrete energy laws which mimic the differential energy law satisfied by the PDE solution. Optimal order error estimates in various norms are established for the numerical solutions of both the semi-discrete and fully discrete methods. Numerical experiments are also presented to show the efficiency of the proposed approach and methods.

math.NA↗

arXiv2009

Finite element methods for a bi-wave equation modeling d-wave superconductors

In this paper we develop two conforming finite element methods for a fourth order bi-wave equation arising as a simplified Ginzburg-Landau-type model for d-wave superconductors in absence of applied magnetic field. Unlike the biharmonic operator $Δ^2$, the bi-wave operator $\Box^2$ is not an elliptic operator, so the energy space for the bi-wave equation is much larger than the energy space for the biharmonic equation. This then makes it possible to construct low order conforming finite elements for the bi-wave equation. However, the existence and construction of such finite elements strongly depends on the mesh. In the paper, we first characterize mesh conditions which allow and not allow construction of low order conforming finite elements for approximating the bi-wave equation. We then construct a cubic and a quartic conforming finite element. It is proved that both elements have the desired approximation properties, and give optimal order error estimates in the energy norm, suboptimal (and optimal in some cases) order error estimates in the $H^1$ and $L^2$ norm. Finally, numerical experiments are presented to guage the efficiency of the proposed finite element methods and to validate the theoretical error bounds.

math.NA↗

arXiv2008

Structural relaxation effects on interface and transport properties of Fe/MgO(001) tunnel junctions

The interface structure of Fe/MgO(100) magnetic tunnel junctions predicted by density functional theory (DFT) depends significantly on the choice of exchange and correlation functional. Bader analysis reveals that structures obtained by relaxing the cell with the local spin-density approximation (LSDA) display a different charge transfer than those relaxed with the generalized gradient approximation (GGA). As a consequence, the electronic transport is found to be extremely sensitive to the interface structure. In particular, the conductance for the LSDA-relaxed geometry is about one order of magnitude smaller than that of the GGA-relaxed one. The high sensitivity of the electronic current to the details of the interface might explain the discrepancy between the experimental and calculated values of magnetoresistance.

cond-mat.mtrl-sci↗

arXiv2008

Discontinuous Galerkin Methods for the Helmholtz Equation with Large Wave Number

This paper develops and analyzes some interior penalty discontinuous Galerkin methods using piecewise linear polynomials for the Helmholtz equation with the first order absorbing boundary condition in the two and three dimensions. It is proved that the proposed discontinuous Galerkin methods are stable (hence well-posed) without any mesh constraint. For each fixed wave number $k$, optimal order (with respect to $h$) error estimate in the broken $H^1$-norm and sub-optimal order estimate in the $L^2$-norm are derived without any mesh constraint. The latter estimate improves to optimal order when the mesh size $h$ is restricted to the preasymptotic regime (i.e., $k^2 h \gtrsim 1$). Numerical experiments are also presented to gauge the theoretical result and to numerically examine the pollution effect (with respect to $k$) in the error bounds. The novelties of the proposed interior penalty discontinuous Galerkin methods include: first, the methods penalize not only the jumps of the function values across the element edges but also the jumps of the normal and tangential derivatives; second, the penalty parameters are taken as complex numbers of positive imaginary parts so essentially and practically no constraint is imposed on the penalty parameters. Since the Helmholtz problem is a non-Hermitian and indefinite linear problem, as expected, the crucial and the most difficult part of the whole analysis is to establish the stability estimates (i.e., a priori estimates) for the numerical solutions. To the end, the cruxes of our analysis are to establish and to make use of a local version of the Rellich identity (for the Laplacian) and to mimic the stability analysis for the PDE solutions given in \cite{cummings00,Cummings_Feng06,hetmaniuk07}.

math.NA↗

arXiv2008

A modified characteristic finite element method for a fully nonlinear formulation of the semigeostrophic flow equations

This paper develops a fully discrete modified characteristic finite element method for a coupled system consisting of the fully nonlinear Monge-Ampére equation and a transport equation. The system is the Eulerian formulation in the dual space for the B. J. Hoskins' semigeostrophic flow equations, which are widely used in meteorology to model slowly varying flows constrained by rotation and stratification. To overcome the difficulty caused by the strong nonlinearity, we first formulate (at the differential level) a vanishing moment approximation of the semigeostrophic flow equations, a methodology recently proposed by the authors \cite{Feng1,Feng2}, which involves approximating the fully nonlinear Monge-Ampére equation by a family of fourth order quasilinear equations. We then construct a fully discrete modified characteristic finite element method for the regularized problem. It is shown that under certain mesh and time stepping constraints, the proposed numerical method converges with an optimal order rate of convergence. In particular, the obtained error bounds show explicit dependence on the regularization parameter $\vepsi$. Numerical tests are also presented to validate the theoretical results and to gauge the efficiency of the proposed fully discrete modified characteristic finite element method.

math.NA↗

arXiv2007

On $p$-harmonic map heat flows for {$1\leq p< \infty$} and their finite element approximations

Motivated by emerging applications from imaging processing, the heat flow of a generalized $p$-harmonic map into spheres is studied for the whole spectrum, $1\leq p<\infty$, in a unified framework. The existence of global weak solutions is established for the flow using the energy method together with a regularization and a penalization technique. In particular, a $BV$-solution concept is introduced and the existence of such a solution is proved for the 1-harmonic map heat flow. The main idea used to develop such a theory is to exploit the properties of measures of the forms $\cA\cdot\nab\bv$ and $\cA\wedge\nab\bv$; which pair a divergence-$L^1$, or a divergence-measure, tensor field $\cA$, and a $BV$-vector field $\bv$. Based on these analytical results, a practical fully discrete finite element method is then proposed for approximating weak solutions of the $p$-harmonic map heat flow, and the convergence of the proposed numerical method is also established.

math.AP↗

arXiv2007

Galerkin Methods for the Fully Nonlinear Monge-Ampère Equation

This paper develops and analyzes finite element Galerkin and spectral Galerkin methods for approximating viscosity solutions of the fully nonlinear Monge-Ampère equation $\det(D^2u^0)=f$ based on the vanishing moment method which was developed by the authors in \cite{Feng2,Feng1}. In this approach, the Monge-Ampère equation is approximated by the fourth order quasilinear equation $-εΔ^2 u^ε+ \det{D^2u^ε} =f$ accompanied by appropriate boundary conditions. This new approach allows one to construct convergent Galerkin numerical methods for the fully nonlinear Monge-Ampère equation, a task which has been impracticable before. In this paper, we first develop some finite element and spectral Galerkin methods for approximating the solution $u^ε$ of the regularized fourth order problem. We then derive optimal order error estimates for the proposed numerical methods. In particular, we track explicitly the dependence of the error bounds on the parameter $\vepsi$, for the error $u^ε-u^ε_h$. Finally, using the Aygris finite element method as an example, we present a detailed numerical study of the rates of convergence in terms of powers of $\vepsi$ for the error $u^0-u_h^\vepsi$, and numerically examine what is the "best" mesh size $h$ in relation to $\vepsi$ in order to achieve these rates.

math.NA↗

arXiv2007

Mixed finite element methods for the fully nonlinear Monge-Ampère equation based on the vanishing moment method

This paper studies mixed finite element approximations of the viscosity solution to the Dirichlet problem for the fully nonlinear Monge-Ampère equation $\det(D^2u^0)=f$ based on the vanishing moment method which was proposed recently by the authors in \cite{Feng2}. In this approach, the second order fully nonlinear Monge-Ampère equation is approximated by the fourth order quasilinear equation $-εΔ^2 u^ε+ \det{D^2u^ε} =f$. It was proved in \cite{Feng1} that the solution $u^ε$ converges to the unique convex viscosity solution $u^0$ of the Dirichlet problem for the Monge-Ampère equation. This result then opens a door for constructing convergent finite element methods for the fully nonlinear second order equations, a task which has been impracticable before. The goal of this paper is threefold. First, we develop a family of Hermann-Miyoshi type mixed finite element methods for approximating the solution $u^ε$ of the regularized fourth order problem, which computes simultaneously $u^\vepsi$ and the moment tensor $σ^\vepsi:=D^2u^ε$. Second, we derive error estimates, which track explicitly the dependence of the error constants on the parameter $\vepsi$, for the errors $u^ε-u^ε_h$ and $σ^\vepsi-σ_h^\vepsi$. Finally, we present a detailed numerical study on the rates of convergence in terms of powers of $\vepsi$ for the error $u^0-u_h^\vepsi$ and $σ^\vepsi-σ_h^\vepsi$, and numerically examine what is the "best" mesh size $h$ in relation to $\vepsi$ in order to achieve these rates.

math.NA↗

arXiv2007

A posteriori error estimates for finite element approximations of the Cahn-Hilliard equation and the Hele-Shaw flow

This paper develops a posteriori error estimates of residual type for conforming and mixed finite element approximations of the fourth order Cahn-Hilliard equation $u_t+\De\bigl(\eps \De u-\eps^{-1} f(u)\bigr)=0$. It is shown that the {\it a posteriori} error bounds depends on $\eps^{-1}$ only in some low polynomial order, instead of exponential order. Using these a posteriori error estimates, we construct an adaptive algorithm for computing the solution of the Cahn-Hilliard equation and its sharp interface limit, the Hele-Shaw flow. Numerical experiments are presented to show the robustness and effectiveness of the new error estimators and the proposed adaptive algorithm.

math.NA↗

arXiv2007

Vanishing moment method and moment solutions for second order fully nonlinear partial differential equations

This paper concerns with numerical approximations of solutions of second order fully nonlinear partial differential equations (PDEs). A new notion of weak solutions, called moment solutions, is introduced for second order fully nonlinear PDEs. Unlike viscosity solutions, moment solutions are defined by a constructive method, called vanishing moment method, hence, they can be readily computed by existing numerical methods such as finite difference, finite element, spectral Galerkin, and discontinuous Galerkin methods with "guaranteed" convergence. The main idea of the proposed vanishing moment method is to approximate a second order fully nonlinear PDE by a higher order, in particular, a fourth order quasilinear PDE. We show by various numerical experiments the viability of the proposed vanishing moment method. All our numerical experiments show the convergence of the vanishing moment method, and they also show that moment solutions coincide with viscosity solutions whenever the latter exist.

math.NA↗