Finite volume element approximation and analysis for a kind of semiconductor device simulation

In this paper, we present a finite volume element scheme for a kind of two dimensional semiconductor device simulation. A general framework is developed for finite volume element approximation of the semiconductor problems. We construct a fully discrete finite volume element scheme based on triangulations with a piecewise linear finite element space and a general type of control volume. Optimal-order convergence in H ¹-norm is derived.     

A coupling method based on new MFE and FE for fourth-order parabolic equation

In this article, a coupling method of new mixed finite element (MFE) and finite element (FE) is proposed and analyzed for fourth-order parabolic partial differential equation. First, the fourth-order parabolic equation is split into the coupled system of second-order equations. Then, an equation is solved by finite element method, the other equation is approximated by the new mixed finite element method, whose flux belongs to the square integrable space replacing the classical H(div;Ω) space. The stability for fully discrete scheme is derived, and both semi-discrete and fully discrete error estimates are obtained. Moreover, the optimal a priori error estimates in L ² and H ¹-norm for both the scalar unknown u and the diffusion term γ and a priori error estimate in (L ²)²-norm for its flux σ are derived. Finally, some numerical results are provided to validate our theoretical analysis.     

A fully discrete spectral method for fractional Cattaneo equation based on Caputo–Fabrizo derivative

Recently Caputo and Fabrizio introduced a new derivative with fractional order without singular kernel. The derivative can be used to describe the material heterogeneities and the fluctuations of different scales. In this article, we derived a new discretization of Caputo–Fabrizio derivative of order α (1 < α < 2) and applied it into the Cattaneo equation. A fully discrete scheme based on finite difference method in time and Legendre spectral approximation in space is proposed. The stability and convergence of the fully discrete scheme are rigorously established. The convergence rate of the fully discrete scheme in H¹ norm is O(τ² + N^1?m), where τ, N and m are the time‐step size, polynomial degree and regularity in the space variable of the exact solution, respectively. Furthermore, the accuracy and applicability of the scheme are confirmed by numerical examples to support the theoretical results.     

Error estimates of CFVE method for fully nonlinear convection‐dominated diffusion problems

Finite volume method and characteristics finite element method are two important methods for solving the partial differential equations. These two methods are combined in this paper to establish a fully discrete characteristics finite volume method for fully nonlinear convection‐dominated diffusion problems. Through detailed theoretical analysis, optimal order H¹ norm error estimates are obtained for this fully discrete scheme. Copyright © 2010 John Wiley & Sons, Ltd.     

Interior penalty discontinuous Galerkin methods with implicit time‐integration techniques for nonlinear parabolic equations

We prove existence and numerical stability of numerical solutions of three fully discrete interior penalty discontinuous Galerkin methods for solving nonlinear parabolic equations. Under some appropriate regularity conditions, we give the l²(H¹) and l^∞(L²) error estimates of the fully discrete symmetric interior penalty discontinuous Galerkin–scheme with the implicit θ ‐schemes in time, which include backward Euler and Crank–Nicolson finite difference approximations. Our estimates are optimal with respect to the mesh size h. The theoretical results are confirmed by some numerical experiments. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Second order unconditionally convergent and energy stable linearized scheme for MHD equations

In this paper, we propose an efficient numerical scheme for magnetohydrodynamics (MHD) equations. This scheme is based on a second order backward difference formula for time derivative terms, extrapolated treatments in linearization for nonlinear terms. Meanwhile, the mixed finite element method is used for spatial discretization. We present that the scheme is unconditionally convergent and energy stable with second order accuracy with respect to time step. The optimal L ² and H ¹ fully discrete error estimates for velocity, magnetic variable and pressure are also demonstrated. A series of numerical tests are carried out to confirm our theoretical results. In addition, the numerical experiments also show the proposed scheme outperforms the other classic second order schemes, such as Crank-Nicolson/Adams-Bashforth scheme, linearized Crank-Nicolson’s scheme and extrapolated Gear’s scheme, in solving high physical parameters MHD problems.     

Average implicit linear difference scheme for generalized Rosenau-Burgers equation

In this paper, a linear three-level average implicit finite difference scheme for the numerical solution of the initial-boundary value problem of Generalized Rosenau-Burgers equation is presented. Existence and uniqueness of numerical solutions are discussed. It is proved that the finite difference scheme is convergent in the order of O(τ² + h²) and stable. Numerical simulations show that the method is efficient.     

Convergence of finite element method for linear second‐order wave equations with discontinuous coefficients

A finite element method is proposed and analyzed for hyperbolic problems with discontinuous coefficients. The main emphasize is given on the convergence of such method. Due to low global regularity of the solutions, the error analysis of the standard finite element method is difficult to adopt for such problems. For a practical finite element discretization, optimal error estimates in L^∞(L²) and L^∞(H¹) norms are established for continuous time discretization. Further, a fully discrete scheme based on a symmetric difference approximation is considered, and optimal order convergence in L^∞(H¹) norm is established. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A new mixed scheme based on variation of constants for Sobolev equation with nonlinear convection term

A new mixed scheme which combines the variation of constants and the H ¹-Galerkin mixed finite element method is constructed for nonlinear Sobolev equation with nonlinear convection term. Optimal error estimates are derived for both semidiscrete and fully discrete schemes. Finally, some numerical results are given to confirm the theoretical analysis of the proposed method.     

and Norms Error Estimates in Finite Element Method for Linear Parabolic Interface Problems

We derive new a priori error estimates for linear parabolic equations with discontinuous coefficients. Due to low global regularity of the solutions the error analysis of the standard finite element method for parabolic problems is difficult to adopt for parabolic interface problems. A finite element procedure is, therefore, proposed and analyzed in this paper. We are able to show that the standard energy technique of finite element method for non-interface parabolic problems can be extended to parabolic interface problems if we allow interface triangles to be curved triangles. Optimal pointwise-in-time error estimates in the L ²(Ω) and H ¹(Ω) norms are shown to hold for the semidiscrete scheme. A fully discrete scheme based on backward Euler method is analyzed and pointwise-in-time error estimates are derived. The interfaces are assumed to be arbitrary shape but smooth for our purpose.     

Stability and Super Convergence Analysis of ADI-FDTD for the 2D Maxwell Equations in a Lossy Medium

Several new energy identities of the two dimensional(2D) Maxwell equations in a lossy medium in the case of the perfectly electric conducting boundary conditions are proposed and proved. These identities show a new kind of energy conservation in the Maxwell system and provide a new energy method to analyze the alternating direction implicit finite difference time domain method for the 2D Maxwell equations (2D-ADI-FDTD). It is proved that 2D-ADI-FDTD is approximately energy conserved, unconditionally stable and second order convergent in the discrete L² and H¹ norms, which implies that 2D-ADI-FDTD is super convergent. By this super convergence, it is simply proved that the error of the divergence of the solution of 2D-ADI-FDTD is second order accurate. It is also proved that the difference scheme of 2D-ADI-FDTD with respect to time t is second order convergent in the discrete H¹ norm. Experimental results to confirm the theoretical analysis on stability, convergence and energy conservation are presented.     

Fully discrete finite element scheme for Maxwell's equations with non-linear boundary condition

We study a full Maxwell's system accompanied with a non-linear degenerate boundary condition, which represents a generalization of the classical Silver-Müller condition for a non-perfect conductor. The relationship between the normal components of electric E and magnetic H field obeys the following power law ν×H=ν×(|E×ν|^α−1E×ν) for some α∈(0,1]. We establish the existence and uniqueness of a weak solution in a suitable function spaces under the minimal regularity assumptions on the boundary Γ and the initial data E₀ and H₀. We design a non-linear time discrete approximation scheme and prove convergence of the approximations to a weak solution. We also derive the error estimates for the time discretization. As a next step we study the fully discrete problem using curl-conforming edge elements and derive the corresponding error estimates. Finally we present some numerical experiments.     

Modified High-order Upwing Method for Convection Diffusion Equation

Abstract   In this paper, we study the high-order upwind finite difference method for steady convection-diffusion problems. Based on the conservative convection-diffusion equation, a high-order upwind finite difference scheme on nonuniform rectangular partition for convection-diffusion equation is proposed. The proposed scheme is in conversation form, satisfies maximum value principle and has second-order error estimates in discrete H ¹ norm. To illustrate our conclusion, several numerical examples are given. Supported by the Research Fund for Doctoral Program of High Education of China State Education Commission.     

Optimal lower bound estimates for the blow-up rate for the Zakharov system in a nonhomogeneous medium

In this paper we obtain lower bound estimates for the blow-up rate of finite time blow-up solutions to the Cauchy problem for the Zakharov system in a nonhomogeneous medium in two space dimensions. By introducing suitable scale transformations of space and time, and the use of compactness arguments, we derive an optimal lower bound estimate in the energy space H²(R²)×L²(R²)×H¹(R²) for the blow-up rate for t near the finite blow-up time T. Also we give an application to the virial identity for the Zakharov system under study.     

New energy identities and super convergence analysis of the energy conserved splitting FDTD methods for 3D Maxwell's equations

The energy‐conserved splitting finite‐difference time‐domain (EC‐S‐FDTD) method has recently been proposed to solve the Maxwell equations with second order accuracy while numerically keep the L² energy conservation laws of the equations. In this paper, the EC‐S‐FDTD scheme for the 3D Maxwell equations is proved to be energy‐conserved and unconditionally stable in the discrete H¹ norm. The EC‐S‐FDTD scheme is of second‐order accuracy both in time step and spatial steps, which suggests the super‐convergence of this scheme in the discrete H¹ norm. And the divergence of the electric field of the EC‐S‐FDTD scheme in the discrete L² norm is second‐order accurate. Numerical experiments confirm our theoretical analysis. Copyright © 2012 John Wiley & Sons, Ltd.     

Error analysis of finite element approximation of a stefan problem with nonlinear free boundary condition

By applying the Landau-type transformation, we transform a Stefan problem with nonlinear free boundary condition into a system consisting of a parabolic equation and the ordinary differential equations. Fully discrete finite element method is developed to approximate the solution of a system of a parabolic equation and the ordinary differential equations. We derive optimal orders of convergence of fully discrete approximations inL2, H¹ and H² normed spaces.     

Error estimates for a finite element-finite volume discretization of convection-diffusion equations

We consider a time-dependent linear convection-diffusion equation. This equation is approximated by a combined finite element-finite volume method: the diffusion term is discretized by Crouzeix-Raviart piecewise linear finite elements, and the convection term by upwind barycentric finite volumes on a triangular grid. An implicit Euler approach is used for time discretization. It is shown that the error associated with this scheme, measured by a discrete L^∞-L²- and L²-H¹-norm, respectively, decays linearly with the mesh size and the time step. This result holds without any link between mesh size and time step. The dependence of the corresponding error bound on the diffusion coefficient is completely explicit.     

Stability and convergence of two‐grid Crank‐Nicolson extrapolation scheme for the time‐dependent natural convection equations

In this paper, we consider the Crank‐Nicolson extrapolation scheme for the 2D/3D unsteady natural convection problem. Our numerical scheme includes the implicit Crank‐Nicolson scheme for linear terms and the recursive linear method for nonlinear terms. Standard Galerkin finite element method is used to approximate the spatial discretization. Stability and optimal error estimates are provided for the numerical solutions. Furthermore, a fully discrete two‐grid Crank‐Nicolson extrapolation scheme is developed, the corresponding stability and convergence results are derived for the approximate solutions. Comparison from aspects of the theoretical results and computational efficiency, the two‐grid Crank‐Nicolson extrapolation scheme has the same order as the one grid method for velocity and temperature in H¹‐norm and for pressure in L²‐norm. However, the two‐grid scheme involves much less work than one grid method. Finally, some numerical examples are provided to verify the established theoretical results and illustrate the performances of the developed numerical schemes.     

Error estimates of H 1-Galerkin mixed finite element method for Schrödinger equation

An H^1-Galerkin mixed finite element method is discussed for a class of second order SchrSdinger equation. Optimal error estimates of semidiscrete schemes are derived for problems in one space dimension. At the same time, optimal error estimates are derived for fully discrete schemes. And it is showed that the H1-Galerkin mixed finite element approximations have the same rate of convergence as in the classical mixed finite element methods without requiring the LBB consistency condition.     

Galerkin methods for the time dependent incompressible elasticity equations

We consider semidiscrete and single step fully discrete approximations to the solutions of the time dependent incompressible elasticity equations. Two methods of dealing with the constraint are analysed: (a) a formulation involving a ‘discrete’ form of incompressibility condition and (b) a penalty formulation. H¹- and L²-error estimates are obtained.