A weak Galerkin finite element method for the second order elliptic problems with mixed boundary conditions

In this paper, a weak Galerkin finite element method is proposed and analyzed for the second-order elliptic equation with mixed boundary conditions. Optimal order error estimates are established in both discrete $H^1$ norm and the standard $L^2$ norm for the corresponding WG approximations. The numerical experiments are presented to verify the efficiency of the method.     

A Sequential Least Squares Method for Elliptic Equations in Non-Divergence Form

We develop a new least squares method for solving the second-order elliptic equations in non-divergence form. Two least-squares-type functionals are proposed for solving the equation in two sequential steps. We first obtain a numerical approximation to the gradient in a piecewise irrotational polynomial space. Then together with the numerical gradient, we seek a numerical solution of the primitive variable in the continuous Lagrange finite element space. The variational setting naturally provides an a posteriori error which can be used in an adaptive refinement algorithm. The error estimates under the $L^2$ norm and the energy norm for both two unknowns are derived. By a series of numerical experiments, we verify the convergence rates and show the efficiency of the adaptive algorithm.     

A Second Order Unconditionally Convergent Finite Element Method for the Thermal Equation with Joule Heating Problem

In this paper, we study the finite element approximation for nonlinear thermal equation. Because the nonlinearity of the equation, our theoretical analysis is based on the error of temporal and spatial discretization. We consider a fully discrete second order backward difference formula based on a finite element method to approximate the temperature and electric potential, and establish optimal $L^2$error estimates for the fully discrete finite element solution without any restriction on the time-step size. The discrete solution is bounded in infinite norm. Finally, several numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.     

Approximate Controllability of the System Governed by Double Coupled Semilinear Degenerate Parabolic Equations

This paper concerns the approximate controllability of the initial-boundary problem of double coupled semilinear degenerate parabolic equations. The equations are degenerate at the boundary, and the control function acts in the interior of the spacial domain and acts only on one equation. We overcome the difficulty of the degeneracy of the equations to show that the problem is approximately controllable in $L^2$ by means of a fixed point theorem and some compact estimates. That is to say, for any initial and desired data in $L^2$, one can find a control function in $L^2$ such that the weak solution to the problem approximately reaches the desired data in $L^2$ at the terminal time.     

油气资源数值模拟的变网格交替方向特征有限元格式和分析

石油科学在有机地球化学、石油的生成、运移、聚集研究取得重大进展,在评价油气资源时,对盆地发育史尤其对流体流动规律和受热变化历史的计算是非常重要的.其数学模型是三维空间非线性偶合偏微分方程组的初边值问题.从实际出发,考虑了流体的压缩性和三维问题大规模科学与工程计算的特征, 提出了一类变网格交替方向特征有限元格式,应用变分形式、算子分裂、广义$L^2$投影、能量方法、负模估计、微分方程先验估计的理论和技巧,得到最佳阶$L^2$误差估计. 此方法已成功应用到油气资源评估数值模拟的生产实践中,成功解决了这一重要问题.     

Exponential Time Differencing-Padé Finite Element Method for Nonlinear Convection-Diffusion-Reaction Equations with Time Constant Delay

In this paper, ETD3-Padé and ETD4-Padé Galerkin finite element methods are proposed and analyzed for nonlinear delayed convection-diffusion-reaction equations with Dirichlet boundary conditions. An ETD-based RK is used for time integration of the corresponding equation. To overcome a well-known difficulty of numerical instability associated with the computation of the exponential operator, the Padé approach is used for such an exponential operator approximation, which in turn leads to the corresponding ETD-Padé schemes. An unconditional $L^2$ numerical stability is proved for the proposed numerical schemes, under a global Lipshitz continuity assumption. In addition, optimal rate error estimates are provided, which gives the convergence order of $O(k^{3}+h^{r})$ (ETD3-Padé) or $O(k^{4}+h^{r})$ (ETD4-Padé) in the $L^2$ norm, respectively. Numerical experiments are presented to demonstrate the robustness of the proposed numerical schemes.     

六阶对流Cahn-Hilliard方程解的整体适定性

In this paper,we consider the global well-posedness of smooth solutions for the Cauchy problem of a sixth order convective Cahn-Hilliard equation with small initial data.We first construct a local smooth solution,then by combining some a priori estimates,continuity argument,the local smooth solution is extended step by step to all t>0 provided that the L1 norm of initial data is suitably small and the smooth nonlinear functions f(u)and g(u)satisfy certain local growth conditions at some fixed point■.     

求解线性Sobolev方程的分裂型最小二乘混合元方法

本文通过引入适当的最小二乘极小化泛函,对一类线性Sobolev方程提出了两种分裂型最小二乘混合元格式,格式最大优点在于将耦合的方程组系统分裂成两个独立的子系统,进而极大降低了原问题求解的难度和规模,理论分析表明格式对原未知量及新引入的未知通量分别具有最优阶L2(Ω)模误差估计和次优阶H(div;Ω)模误差估计.数值试验很好的验证了这一点.     

An Energy Stable Filtered Backward Euler Scheme for the MBE Equation with Slope Selection

As a promising strategy to adjust the order in the variable-order BDF algorithm, a time filtered backward Euler scheme is investigated for the molecular beam epitaxial equation with slope selection. The temporal second-order convergence in the $L^2$ norm is established under a convergence-solvability-stability (CSS)-consistent time-step constraint. The CSS-consistent condition means that the maximum step-size limit required for convergence is of the same order to that for solvability and stability (in certain norms) as the small interface parameter $ε → 0^+.$ Similar to the backward Euler scheme, the time filtered backward Euler scheme preserves some physical properties of the original problem at the discrete levels, including the volume conservation, the energy dissipation law and $L^2$ norm boundedness. Numerical tests are included to support the theoretical results.     

Existence and Decay of Global Strong Solution to 3D Density-Dependent Boussinesq Equations with Vacuum

This paper is concerned with the initial boundary problem for the three-dimensional density-dependent Boussinesq equations with vacuum. We obtain the existence of the global strong solution under the initial density in the norm $L^{\infty}$ is small enough without any smallness condition of $u$ and $\theta$. Furthermore, the exponential decay rates of the solution and their derivatives in some norm was established. In addition, we show that the solution and their derivatives are monotonically decreasing with respect to time $t$ on $[0,T]$.     

Finite difference/$H^1$-Galerkin MFE procedure for a fractional water wave model

In this article, an $H^1$-Galerkin mixed finite element (MFE) method for solving the time fractional water wave model is presented. First-order backward Euler difference method and $L1$ formula are applied to approximate integer derivative and Caputo fractional derivative with order $1/2$, respectively, and $H^1$-Galerkin mixed finite element method is used to approximate the spatial direction. The analysis of stability for fully discrete mixed finite element scheme is made and the optimal space-time orders of convergence for two unknown variables in both $H^1$-norm and $L^2$-norm are derived. Further, some computing results for a priori analysis and numerical figures based on four changed parameters in the studied problem are given to illustrate the effectiveness of the current method     

Energy Identities and Stability Analysis of the Yee Scheme for 3D Maxwell Equations

In this paper numerical energy identities of the Yee scheme on uniform grids for three dimensional Maxwell equations with periodic boundary conditions are proposed and expressed in terms of the $L^2$, $H^1$ and $H^2$ norms. The relations between the $H^1$ or $H^2$ semi-norms and the magnitudes of the curls or the second curls of the fields in the Yee scheme are derived. By the $L^2$ form of the identity it is shown that the solution fields of the Yee scheme is approximately energy conserved. By the $H^1$ or $H^2$ semi norm of the identities, it is proved that the curls or the second curls of the solution of the Yee scheme are approximately magnitude (or energy)-conserved. From these numerical energy identities, the Courant-Friedrichs-Lewy (CFL) stability condition is re-derived, and the stability of the Yee scheme in the $L^2$, $H^1$ and $H^2$ norms is then proved. Numerical experiments to compute the numerical energies and convergence orders in the $L^2$, $H^1$ and $H^2$ norms are carried out and the computational results confirm the analysis of the Yee scheme on energy conservation and stability analysis.     

Convergence of Weak Galerkin Finite Element Method for Second Order Linear Wave Equation in Heterogeneous Media

Weak Galerkin finite element method is introduced for solving wave equation with interface on weak Galerkin finite element space $(\mathcal{P}_k(K), \mathcal{P}_{k−1}(∂K), [\mathcal{P}_{k−1}(K)]^2).$ Optimal order a priori error estimates for both space-discrete scheme and implicit fully discrete scheme are derived in $L^∞(L^2)$ norm. This method uses totally discontinuous functions in approximation space and allows the usage of finite element partitions consisting of general polygonal meshes. Finite element algorithm presented here can contribute to a variety of hyperbolic problems where physical domain consists of heterogeneous media.     

Two-level iteration penalty and variational multiscale method for steady incompressible flows

In this paper, we study two-level iteration penalty and variational multiscale method for the approximation of steady Navier-Stokes equations at high Reynolds number. Comparing with classical penalty method, this new method does not require very small penalty parameter $\varepsilon$. Moreover, two-level mesh method can save a large amount of CPU time. The error estimates in $H^1$ norm for velocity and in $L^2$ norm for pressure are derived. Finally, two numerical experiments are shown to support the efficiency of this new method.     

-framework for continuous dependence and error estimates for quasilinear anisotropic degenerate parabolic equations

We develop a general -framework for deriving continuous dependence and error estimates for quasilinear anisotropic degenerate parabolic equations with the aid of the Chen-Perthame kinetic approach. We apply our -framework to establish an explicit estimate for continuous dependence on the nonlinearities and an optimal error estimate for the vanishing anisotropic viscosity method, without imposition of bounded variation of the approximate solutions. Finally, as an example of a direct application of this framework to numerical methods, we focus on a linear convection-diffusion model equation and derive an error estimate for an upwind-central finite difference scheme.

     

弱奇异时滞Volterra积分方程雅可比收敛分析

本文利用雅可比谱配置方法研究弱奇异时滞Volterra积分方程,分别得到真解与近似解在L∞和L2ω-μ,0范数意义下呈现指数收敛的结论,数值仿真结果验证理论分析的正确性.     

Stability and Convergence Analyses of the FDM Based on Some L-Type Formulae for Solving the Subdiffusion Equation

Some well-known L-type formulae, i.e., L1, L1-2, and L1-2-3 formulae, are usually employed to approximate the Caputo fractional derivative of order α ∈ (0, 1). In this paper, we aim to elaborate on the stability and convergence analyses of some finite difference methods (FDMs) for solving the subdiffusion equation, i.e., a diffusion equation which exploits the Caputo time-fractional derivative of order $α$. In fact, the FDMs considered here are based on the usual central difference scheme for the spatial derivative, and the Caputo derivative is approximated by using methods such as the L1, L1-2, and L1-2-3 formulae. Thanks to a specific type of the discrete version of the Gronwall inequality, we show that the FDMs are unconditionally stable in the maximum norm and also discrete $H^1$ norm. Then, we prove that the finite difference method which uses the L1, L1-2, and L1-2-3 formulae has the global order of convergence $2−α$, $3−α$, and 3, respectively. Finally, some numerical tests confirm the theoretical results. A brief conclusion finishes the paper.     

Alternating Direction Implicit Galerkin Finite Element Method for the Two-Dimensional Time Fractional Evolution Equation

New numerical techniques are presented for the solution of the two-dimensional time fractional evolution equation in the unit square. In these methods, Galerkin finite element is used for the spatial discretization, and, for the time stepping, new alternating direction implicit (ADI) method based on the backward Euler method combined with the first order convolution quadrature approximating the integral term are considered. The ADI Galerkin finite element method is proved to be convergent in time and in the $L^2$ norm in space. The convergence order is$\mathcal{O}$($k$|ln $k$|+$h^r$), where $k$ is the temporal grid size and $h$ is spatial grid size in the $x$ and $y$ directions, respectively. Numerical results are presented to support our theoretical analysis.     

Existence and Uniqueness of Smooth Solution for a Four-waves Coupled System

In this paper, we consider a four-waves coupled system which describes the interaction between particles. Based on the uniform bound and strong convergence property in lower order norm, local existence and uniqueness of smooth solution is established by a limiting argument. Moreover, we show the solution exists globally in two dimensional case under certain condition on the size for $L^2$ norm of the initial data.     

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 $h_\T$.