An Energy-Stable Parametric Finite Element Method for Simulating Solid-State Dewetting Problems in Three Dimensions

We propose an accurate and energy-stable parametric finite element method for solving the sharp-interface continuum model of solid-state dewetting in three-dimensional space. The model describes the motion of the film\slash vapor interface with contact line migration and is governed by the surface diffusion equation with proper boundary conditions at the contact line. We present a weak formulation for the problem, in which the contact angle condition is weakly enforced. By using piecewise linear elements in space and backward Euler method in time, we then discretize the formulation to obtain a parametric finite element approximation, where the interface and its contact line are evolved simultaneously. The resulting numerical method is shown to be well-posed and unconditionally energy-stable. Furthermore, the numerical method is generalized to the case of anisotropic surface energies in the Riemannian metric form. Numerical results are reported to show the convergence and efficiency of the proposed numerical method as well as the anisotropic effects on the morphological evolution of thin films in solid-state dewetting.     

Numerical Analysis of the Allen-Cahn Equation with Coarse Meshes

In this paper, we consider the finite difference semi-discretization of the Allen-Cahn equation with the diffuse interface parameter $\varepsilon$. While it is natural to make the mesh size parameter $h$ smaller than $\varepsilon$, it is desirable that $h$ is as big as possible in view of computational costs. In fact, when $h$ is bigger than $\varepsilon$ (i.e., the mesh is relatively coarse), it is observed that the numerical solution does not move at all. The purpose of this paper is to clarify the mechanism of this phenomenon. We will prove that the numerical solution converges to that of the ordinary equation without the diffusion term if $h$ is bigger than $\varepsilon$. Numerical examples are presented to support the result.     

On the tangential Cauchy‐Fueter operators on nondegenerate quadratic hypersurfaces in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathbb {H}^2}$\end{document}

On quadratic hypersurfaces in $\mathbb {H}^2$, we find the explicit forms of tangential Cauchy‐Fueter operators and associated tangential Laplacians □_b. Then by using the Fourier transformation on the associated nilpotent Lie groups of step two, we construct the relative fundamental solutions to the tangential Laplacians and Szegö kernels on the nondegenerate quadratic hypersurfaces. It is different from the complex case that the quaternionic tangential structures on the nondegenerate quadratic hypersurfaces in $\mathbb {H}^2$ cannot be reduced to one standard model and the non‐homogeneous tangential Cauchy‐Fueter equations are solvable even in many convex cases.     

ERROR ANALYSIS OF A STABILIZED FINITE ELEMENT METHOD FOR THE GENERALIZED STOKES PROBLEM

This paper is devoted to the establishment of sharper $a$ $priori$stability and error estimates of a stabilized finite element method proposed by Barrenechea and Valentin for solving the generalized Stokes problem, which involves a viscosity $\nu$ and a reaction constant $\sigma$. With the establishment of sharper stability estimates and the help of $ad$ $hoc$finite element projections, we can explicitly establish the dependence of error bounds of velocity and pressure on the viscosity $\nu$, the reaction constant $\sigma$, and the mesh size $h$. Our analysis reveals that the viscosity $\nu$ and the reaction constant $\sigma$ respectively act in the numerator position and the denominator position in the error estimates of velocity and pressure in standard norms without any weights. Consequently, the stabilization method is indeed suitable for the generalized Stokes problem with a small viscosity $\nu$ and a large reaction constant $\sigma$. The sharper error estimates agree very well with the numerical results.     

Analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interface limits

We propose and analyze a fully discrete finite element scheme for the phase field model describing the solidification process in materials science. The primary goal of this paper is to establish some useful a priori error estimates for the proposed numerical method, in particular, by focusing on the dependence of the error bounds on the parameter , known as the measure of the interface thickness. Optimal order error bounds are shown for the fully discrete scheme under some reasonable constraints on the mesh size and the time step size . In particular, it is shown that all error bounds depend on only in some lower polynomial order for small . The cruxes of the analysis are to establish stability estimates for the discrete solutions, to use a spectrum estimate result of Chen, and to establish a discrete counterpart of it for a linearized phase field operator to handle the nonlinear effect. Finally, as a nontrivial byproduct, the error estimates are used to establish convergence of the solution of the fully discrete scheme to solutions of the sharp interface limits of the phase field model under different scaling in its coefficients. The sharp interface limits include the classical Stefan problem, the generalized Stefan problems with surface tension and surface kinetics, the motion by mean curvature flow, and the Hele-Shaw model.

     

The Finite Element Analysis of the Controlled-Source Electromagnetic Induction Problems by Fractional-Step Projection Method

This paper provides a convergence analysis of a fractional-step projection method for the controlled-source electromagnetic induction problems in heterogenous electrically conduting media by means of finite element approximations. Error estimates in finite time are given. And it is verified that provided the time step $\tau$ is sufficiently small, the proposed algorithm yields for finite time $T$ an error of $\mathcal{O}(h^s+\tau)$) in the $L^2$-norm for the magnetic field $\boldsymbol{H},$ where $h$ is the mesh size and $1/2 < s≤1$.     

拟矩形剖分上的样条空间的维数

矩形剖分~(记为$\Delta_{QR}$)~是指在矩形剖分~(记为$\Delta_{R}$)的基础上进行局部修改后得到的剖分,通常包括T-剖分~(记为$\Delta_{T}$)~和L-剖分~(记为$\Delta_{L}$).本文利用光滑余因子协调方法讨论了该剖分上的二元样条空间$S^\mu_k(\Delta_{QR})$的维数.在满足一定约束条件下, 得到了仅依赖于样条空间的次数,光滑度和剖分拓扑结构的显式维数公式.     

A Two-Grid Finite Element Approximation for Nonlinear Time Fractional Two-Term Mixed Sub-Diffusion and Diffusion Wave Equations

In this paper, we develop a two-grid method (TGM) based on the FEM for 2D nonlinear time fractional two-term mixed sub-diffusion and diffusion wave equations. A two-grid algorithm is proposed for solving the nonlinear system, which consists of two steps: a nonlinear FE system is solved on a coarse grid, then the linearized FE system is solved on the fine grid by Newton iteration based on the coarse solution. The fully discrete numerical approximation is analyzed, where the Galerkin finite element method for the space derivatives and the finite difference scheme for the time Caputo derivative with order $\alpha\in(1,2)$ and $\alpha_{1}\in(0,1)$. Numerical stability and optimal error estimate $O(h^{r+1}+H^{2r+2}+\tau^{\min\{3-\alpha,2-\alpha_{1}\}})$ in $L^{2}$-norm are presented for two-grid scheme, where $t,$ $H$ and $h$ are the time step size, coarse grid mesh size and fine grid mesh size, respectively. Finally, numerical experiments are provided to confirm our theoretical results and effectiveness of the proposed algorithm.     

CONVERGENCE OF AN ALTERNATING A-φ SCHEME FOR QUASI-MAGNETOSTATIC EDDY CURRENT PROBLEM

We propose in this paper an alternating A-$\phi$ method for the quasi-magnetostatic eddy current problem by means of finite element approximations. Bounds for continuous and discrete error in finite time are given. And it is verified that provided the time step $\tau$ is sufficiently small, the proposed algorithm yields for finite time $T$ an error of $O(h+\tau^{1/2})$ in the $L^2$-norm for the magnetic field $H(= \mu^{-1} \nabla \times A)$, where $h$ is the mesh size, $\mu$ the magnetic permeability.     

A Fast Conservative Scheme for the Space Fractional Nonlinear Schrödinger Equation with Wave Operator

A new efficient compact difference scheme is proposed for solving a space fractional nonlinear Schrödinger equation with wave operator. The scheme is proved to conserve the total mass and total energy in a discrete sense. Using the energy method, the proposed scheme is proved to be unconditionally stable and its convergence order is shown to be of $ \mathcal{O}( h^6 + \tau^2) $ in the discrete $ L_2 $ norm with mesh size $ h $ and the time step $ \tau $. Moreover, a fast difference solver is developed to speed up the numerical computation of the scheme. Numerical experiments are given to support the theoretical analysis and to verify the efficiency, accuracy, and discrete conservation laws.     

On traces for functional spaces related to Maxwell's equations Part II: Hodge decompositions on the boundary of Lipschitz polyhedra and applications

Hodge decompositions of tangential vector fields defined on piecewise regular manifolds are provided. The first step is the study of L² tangential fields and then the attention is focused on some particular Sobolev spaces of order $‐{1\over 2}$\nopagenumbers\end . In order to reach this goal, it is required to properly define the first order differential operators and to investigate their properties. When the manifold Γ is the boundary of a polyhedron Ω, these spaces are important in the analysis of tangential trace mappings for vector fields in H ( curl , Ω) on the whole boundary or on a part of it. By means of these Hodge decompositions, one can then provide a complete characterization of these trace mappings: general extension theorems, from the boundary, or from a part of it, to the inside; definition of suitable dualities and validity of integration by parts formulae. Copyright © 2001 John Wiley & Sons, Ltd.     

An Adaptive Mesh-Refining Algorithm Allowing for an <Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript> Stable <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> Projection onto Courant Finite Element Spaces

Suppose $\cal{S}^1({\cal T})\subset H^1(\Omega)$ is the $P_1$-finite element space of $\cal{T}$-piecewise affine functions based on a regular triangulation $\cal{T}$ of a two-dimensional surface $\Omega$ into triangles. The $L^2$ projection $\Pi$ onto $\cal{S}^1(\cal{T})$ is $H^1$ stable if $\norm{\Pi v}{H^1(\Omega)}\le C\norm{v}{H^1(\Omega)}$ for all $v$ in the Sobolev space $H^1(\Omega)$ and if the bound $C$ does not depend on the mesh-size in $\cal{T}$ or on the dimension of $\cal{S}^1(\cal{T})$. \hskip 1em A red–green–blue refining adaptive algorithm is designed which refines a coarse mesh $\cal{T}_0$ successively such that each triangle is divided into one, two, three, or four subtriangles. This is the newest vertex bisection supplemented with possible red refinements based on a careful initialization. The resulting finite element space allows for an $H^1$ stable $L^2$ projection. The stability bound $C$ depends only on the coarse mesh $\cal{T}_0$ through the number of unknowns, the shapes of the triangles in $\cal{T}_0$, and possible Dirichlet boundary conditions. Our arguments also provide a discrete version $\norm{h_\cal{T}^{-1}\,\Pi v}{L^2(\Omega)}\le C\norm{h_\cal{T}^{-1}\,v}{L^2(\Omega)}$ in $L^2$ norms weighted with the mesh-size $h_\T$.     

Robust a Simulation for Shallow Flows with Friction on Rough Topography

In this paper, we propose a robust finite volume scheme to numerically solve the shallow water equations on complex rough topography. The major difficulty of this problem is introduced by the stiff friction force term and the wet/dry interface tracking. An analytical integration method is presented for the friction force term to remove the stiffness. In the vicinity of wet/dry interface, the numerical stability can be attained by introducing an empirical parameter, the water depth tolerance, as extensively adopted in literatures. We propose a problem independent formulation for this parameter, which provides a stable scheme and preserves the overall truncation error of $\mathbb{O}$∆$x^3$. The method is applied to solve problems with complex rough topography, coupled with $h$-adaptive mesh techniques to demonstrate its robustness and efficiency.     

Extrapolation of Nyström Solutions of Boundary Integral Equations on Non-Smooth Domains

The interior Dirichlet problem for Laplace's equation on a plane polygonal region $\Omega$ with boundary $\Gamma$ may be reformulated as a second kind integral equation on $\Gamma$. This equation may be solved by the Nyström method using the composite trapezoidal rule. It is known that if the mesh has $O(n)$ points and is graded appropriately, then $O(1/n^2)$ convergence is obtained for the solution of the integral equation and the associated solution to the Dirichlet problem at any $x\in \Omega$. We present a simple extrapolation scheme which increases these rates of convergence to $O(1/n^4)$ .     

Optimal anisotropic meshes for minimizing interpolation errors in -norm

In this paper, we present a new optimal interpolation error estimate in norm ( ) for finite element simplicial meshes in any spatial dimension. A sufficient condition for a mesh to be nearly optimal is that it is quasi-uniform under a new metric defined by a modified Hessian matrix of the function to be interpolated. We also give new functionals for the global moving mesh method and obtain optimal monitor functions from the viewpoint of minimizing interpolation error in the norm. Some numerical examples are also given to support the theoretical estimates.

     

Surface Approximation Is Sometimes Easier than Surface Integration

The approximation and integration problems consist of finding anapproximation to a function $f$ or its integral over some fixeddomain $\Sigma$. For the classical version of these problems, wehave partial information about the functions $f$ and completeinformation about the domain $\Sigma$; for example, $\Sigma$ might bea cube or ball in $\reals^d$. When this holds, it is generally thecase that integration is not harder than approximation; moreover,integration can be much easier than approximation. What happens if wehave partial information about $\Sigma$? This paper studies thesurface approximation and surface integration problems, in which$\Sigma=\Sigma_g$ for functions $g$. More specifically, thefunctions $f$ are $r$ times continuously differentiable scalarfunctions of $l$ variables, and the functions $g$ are $s$ times continuouslydifferentiable injective functions of $d$ variables with$l$ components. The class of surfaces considered is generated as images of cubes or balls, or as oriented cellulated regions.Error for the surface approximation problem is measured in the $L_q$-sense.These problems are well defined, provided that $d\le l$, $r\ge 0$, and$s\ge 1$. Information consists of function evaluations of $f$ and $g$.We show that the $\e$-complexity of surfaceapproximation is proportional to $(1/\e)^{1/\mu}$ with $\mu=\mrs/d$.We also show that if $s\ge 2$, then the $\e$-complexity of surfaceintegration is proportional to $(1/\e)^{1/\nu}$ with$\nu=\min\left\{\frac{r}{d},\frac{s-\delta_{s,1}(1-\delta_{d,l})}{\min\{d,l-1\}}\right\}.$(This bound holds as well for several subcases of $s=1$; we conjecturethat it holds for all $r\ge0$, $s\ge1$, and $d\le l$.) Using theseresults, we determine when surface approximation is easier than, aseasy as, or harder than, surface integration; all three possibilitiescan occur. In particular, we find that if $r=s=1$ and $d相似文献   

REGULARITY ESTIMATES FOR THE OBLIQUE DERIVATIVE PROBLEM ON NON-SMOOTH DOMAINS （I）

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The stability of modified Runge-Kutta methods for the pantograph equation

In the present paper, the modified Runge-Kutta method is constructed, and it is proved that the modified Runge-Kutta method preserves the order of accuracy of the original one. The necessary and sufficient conditions under which the modified Runge-Kutta methods with the variable mesh are asymptotically stable are given. As a result, the -methods with , the odd stage Gauss-Legendre methods and the even stage Lobatto IIIA and IIIB methods are asymptotically stable. Some experiments are given.

     

A new family of stable mixed finite elements for the 3D Stokes equations

A natural mixed-element approach for the Stokes equations in the velocity-pressure formulation would approximate the velocity by continuous piecewise-polynomials and would approximate the pressure by discontinuous piecewise-polynomials of one degree lower. However, many such elements are unstable in 2D and 3D. This paper is devoted to proving that the mixed finite elements of this - type when satisfy the stability condition--the Babuska-Brezzi inequality on macro-tetrahedra meshes where each big tetrahedron is subdivided into four subtetrahedra. This type of mesh simplifies the implementation since it has no restrictions on the initial mesh. The new element also suits the multigrid method.