Two-grid methods for semi-linear elliptic interface problems by immersed finite element methods

In this paper,two-grid immersed finite element (IFE) algorithms are proposed and analyzed for semi-linear interface problems with discontinuous diffusion coefficients in two dimension.Because of the advantages of finite element (FE) formulation and the simple structure of Cartesian grids,the IFE discretization is used in this paper.Two-grid schemes are formulated to linearize the FE equations.It is theoretically and numerically illustrated that the coarse space can be selected as coarse as H =O(h^1/4)(or H =O(h^1/8)),and the asymptotically optimal approximation can be achieved as the nonlinear schemes.As a result,we can settle a great majority of nonlinear equations as easy as linearized problems.In order to estimate the present two-grid algorithms,we derive the optimal error estimates of the IFE solution in the L^p norm.Numerical experiments are given to verify the theorems and indicate that the present two-grid algorithms can greatly improve the computing efficiency.

Local and parallel finite element algorithms based on two-grid discretization for steady Navier-Stokes equations

Local and parallel finite element algorithms based on two-grid discretization for Navier-Stokes equations in two dimension are presented. Its basis is a coarse finite element space on the global domain and a fine finite element space on the subdomain. The local algorithm consists of finding a solution for a given nonlinear problem in the coarse finite element space and a solution for a linear problem in the fine finite element space, then droping the coarse solution of the region near the boundary. By overlapping domain decomposition, the parallel algorithms are obtained. This paper analyzes the error of these algorithms and gets some error estimates which are better than those of the standard finite element method. The numerical experiments are given too. By analyzing and comparing these results, it is shown that these algorithms are correct and high efficient.

OPTIMAL ERROR ESTIMATES OF A DECOUPLED SCHEME BASED ON TWO-GRID FINITE ELEMENT FOR MIXED NAVIER-STOKES/DARCY MODEL

Although the two-grid finite element decoupled scheme for mixed Navier-Stokes/ Darcy model in literatures has given the numerical results of optimal convergence order, the theoretical analysis only obtain the optimal error order for the porous media flow and a non-optimal error order for the fluid flow. In this article, we give a more rigorous of the error analysis for the fluid flow and obtain the optimal error estimates of the velocity and the pressure.

A two-grid algorithm based on Newton iteration for the stream function form of the Navier-Stokes equations

In this paper, we propose a two-grid algorithm for solving the stream function formulation of the stationary Navies-Stokes equations. The algorithm is constructed by reducing the original system to one small, nonlinear system on the coarse mesh space and two similar linear systems （with same stiffness matrix but different right-hand side） on the fine mesh space. The convergence analysis and error estimation of the algorithm are given for the case of conforming elements. Furthermore, the Mgorithm produces a numerical solution with the optimal asymptotic H^2-error. Finally, we give a numerical illustration to demonstrate the effectiveness of the two-grid algorithm for solving the Navier-Stokes equations.

Modified two-grid method for solving coupled Navier-Stokes/Darcy model based on Newton iteration

A new decoupled two-gird algorithm with the Newton iteration is proposed for solving the coupled Navier-Stokes/Darcy model which describes a fluid flow filtrating through porous media. Moreover the error estimate is given, which shows that the same order of accuracy can be achieved as solving the system directly in the fine mesh when h = H2. Both theoretical analysis and numerical experiments illustrate the efficiency of the algorithm for solving the coupled problem.

TWO-GRID METHOD FOR CHARACTERISTICS FINITE-ELEMENT SOLUTION OF 2D NONLINEAR CONVECTION-DOMINATED DIFFUSION PROBLEM

For two-dimension nonlinear convection diffusion equation, a two-grid method of characteristics finite-element solution was constructed. In this method the nonlinear iterations is only to execute on the coarse grid and the fine-grid solution can be obtained in a single linear step. For the nonlinear convection-dominated diffusion equation, this method can not only stabilize the numerical oscillation but also accelerate the convergence and improve the computational efficiency. The error analysis demonstrates if the mesh sizes between coarse-grid and fine-grid satisfy the certain relationship, the two-grid solution and the characteristics finite-element solution have the same order of accuracy. The numerical is more efficient than that of characteristics example confirms that the two-grid method finite-element method.

Two-grid partition of unity method for second order elliptic problems

A two-grid partition of unity method for second order elliptic problems is proposed and analyzed. The standard two-grid method is a local and parallel method usually leading to a discontinuous solution in the entire computational domain. Partition of unity method is employed to glue all the local solutions together to get the global continuous one, which is optimal in HI-norm. Furthermore, it is shown that the L^2 error can be improved by using the coarse grid correction. Numerical experiments are reported to support the theoretical results.

Analysis of a two-grid method for semiconductor device problem

The mathematical model of a semiconductor device is governed by a system of quasi-linear partial differential equations.The electric potential equation is approximated by a mixed finite element method,and the concentration equations are approximated by a standard Galerkin method.We estimate the error of the numerical solutions in the sense of the Lqnorm.To linearize the full discrete scheme of the problem,we present an efficient two-grid method based on the idea of Newton iteration.The main procedures are to solve the small scaled nonlinear equations on the coarse grid and then deal with the linear equations on the fine grid.Error estimation for the two-grid solutions is analyzed in detail.It is shown that this method still achieves asymptotically optimal approximations as long as a mesh size satisfies H=O(h^1/2).Numerical experiments are given to illustrate the efficiency of the two-grid method.

DEM-CFD simulation of modular PB-FHR core with two-grid method

For designing and optimizing the reactor core of modular pebble-bed fluoride salt-cooled high-temperature reactor(PB-FHR),it is of importance to simulate the coupled fluid and particle flow due to strong coolantpebble interactions.Computational fluid dynamics and discrete element method(DEM) coupling approach can be used to track particles individually while it requires a fluid cell being greater than the pebble diameter.However,the large size of pebbles makes the fluid grid too coarse to capture the complicated flow pattern.To solve this problem,a two-grid approach is proposed to calculate interphase momentum transfer between pebbles and coolant without the constraint on the shape and size of fluid meshes.The solid velocity,fluid velocity,fluid pressure and void fraction are mapped between hexahedral coarse particle grid and tine fluid grid.Then the total interphase force can be calculated independently to speed up computation.To evaluate suitability of this two-grid approach,the pressure drop and minimum fluidization velocity of a fluidized bed were predicted,and movements of the pebbles in complex flow field were studied experimentally and numerically.The spouting fluid through a central inlet pipe of a scaled visible PB-FHR core facility was set up to provide the complex flow field.Water was chosen as Liquid to simulate the molten salt coolant,and polypropylene balls were used to simulate the pebble fuels.Results show that the pebble flow pattern captured from experiment agrees well with the simulation from two-grid approach,hence the applicability of the two-grid approach for the later PB-FHR core design.

Two-grid method for characteristic mixed finite-element solutions of nonlinear convection-diffusion equations

A two-grid method for solving nonlinear convection-dominated diffusion equations is presented. The method use discretizations based on a characteristic mixed finite-element method and give the linearization for nonlinear systems by two steps. The error analysis shows that the two-grid scheme combined with the characteristic mixed finite-element method can decrease numerical oscillation caused by dominated convections and solve nonlinear advection-dominated diffusion problems efficiently.

A Two-Grid Technique for the Penalty Method of the Steady Navier-Stokes Equations

A two grid technique for solving the steady incompressible Navier Stokes equations in a penalty method was presented and the convergence of numerical solutions was analyzed. If a coarse size H and a fine size h satisfy H=O(h 13-s )(s=0(n=2);s=12(n=3), where n is a space dimension), this method has the same convergence accuracy as the usual finite element method. But the two grid method can save a lot of computation time for its brief calculation. Moreover, a numerical test was couducted in order to verify the correctness of above theoretical analysis.

UNIFORM SUPERCONVERGENCE ANALYSIS OF A TWO-GRID MIXED FINITE ELEMENT METHOD FOR THE TIME-DEPENDENT BI-WAVE PROBLEM MODELING D-WAVE SUPERCONDUCTORS

In this paper,a two-grid mixed finite element method(MFEM)of implicit Backward Euler(BE)formula is presented for the fourth order time-dependent singularly perturbed Bi-wave problem for d-wave superconductors by the nonconforming EQ_(1)^(rot) element.In this approach,the original nonlinear system is solved on the coarse mesh through the Newton iteration method,and then the linear system is computed on the fine mesh with Taylor’s expansion.Based on the high accuracy results of the chosen element,the uniform superclose and superconvergent estimates in the broken H^(1)-norm are derived,which are independent of the negative powers of the perturbation parameter appeared in the considered problem.Numerical results illustrate that the computing cost of the proposed two-grid method is much less than that of the conventional Galerkin MFEM without loss of accuracy.

THE NONCONFORMING CROUZEIX-RAVIART ELEMENT APPROXIMATION AND TWO-GRID DISCRETIZATIONS FOR THE ELASTICEIGENVALUE PROBLEM

In this paper,we extend the work of Brenner and Sung[Math.Comp.59,321–338(1992)]and present a regularity estimate for the elastic equations in concave domains.Based on the regularity estimate we prove that the constants in the error estimates of the nonconforming Crouzeix-Raviart element approximations for the elastic equations/eigenvalue problem are independent of Laméconstant,which means the nonconforming Crouzeix-Raviart element approximations are locking-free.We also establish two kinds of two-grid discretization schemes for the elastic eigenvalue problem,and analyze that when the mesh sizes of coarse grid and fine grid satisfy some relationship,the resulting solutions can achieve the optimal accuracy.Numerical examples are provided to show the efficiency of two-grid schemes for the elastic eigenvalue problem.

准噶尔盆地深层油气勘探地震采集关键技术及效果

准噶尔盆地目前4 500 m以深的深层油气探明率较低,近年来深层油气勘探目标成为该地区勘探发现的战略接替区。为了查清地下深层结构,重新构建和部署了准噶尔盆地二维地震格架线,利用以往采集资料和重新采集的新资料拼接形成了数十条线组成的二维格架网。基于“两宽一高”技术体系,采用一系列采集关键技术,如“高激发密度、高接收密度、高覆盖密度和长排列”的“三高一长”采集技术,解决了准噶尔盆地深层油气勘探地震采集面临的深层目标地震信号“能量低、干扰强、成像差和不成像”等难题。新采集并经后期处理后的资料成像效果显著提高,为建立盆地石炭系-二叠系统一的地层格架、明确石炭系隆凹格局、查清富烃凹陷下组合地层及烃源岩展布特征提供了良好的资料基础,为下一步深层油气勘探工作提供了有效指导。

TWO-GRID DISCRETIZATION SCHEMES OF THE NONCONFORMING FEM FOR EIGENVALUE PROBLEMS

This paper extends the two-grid discretization scheme of the conforming finite elements proposed by Xu and Zhou （Math. Comput., 70 （2001）, pp.17-25） to the nonconforming finite elements for eigenvalue problems. In particular, two two-grid discretization schemes based on Rayleigh quotient technique are proposed. By using these new schemes, the solution of an eigenvalue problem on a fine mesh is reduced to that on a much coarser mesh together with the solution of a linear algebraic system on the fine mesh. The resulting solution still maintains an asymptotically optimal accuracy. Comparing with the two-grid discretization scheme of the conforming finite elements, the main advantages of our new schemes are twofold when the mesh size is small enough. First, the lower bounds of the exact eigenvalues in our two-grid discretization schemes can be obtained. Second, the first eigenvalue given by the new schemes has much better accuracy than that obtained by solving the eigenvalue problems on the fine mesh directly.

Generalized Rayleigh quotient and finite element two-grid discretization schemes

This study discusses generalized Rayleigh quotient and high efficiency finite element discretization schemes. Some results are as follows: 1) Rayleigh quotient accelerate technique is extended to nonselfadjoint problems. Generalized Rayleigh quotients of operator form and weak form are defined and the basic relationship between approximate eigenfunction and its generalized Rayleigh quotient is established. 2) New error estimates are obtained by replacing the ascent of exact eigenvalue with the ascent of finite element approximate eigenvalue. 3) Based on the work of Xu Jinchao and Zhou Aihui, finite element two-grid discretization schemes are established to solve nonselfadjoint elliptic differential operator eigenvalue problems and these schemes are used in both conforming finite element and non-conforming finite element. Besides, the efficiency of the schemes is proved by both theoretical analysis and numerical experiments. 4) Iterated Galerkin method, interpolated correction method and gradient recovery for selfadjoint elliptic differential operator eigenvalue problems are extended to nonselfadjoint elliptic differential operator eigenvalue problems.

A TWO-GRID FINITE-ELEMENT METHOD FOR THE NONLINEAR SCHRODINGER EQUATION

In this paper, some two-grid finite element schemes are constructed for solving the nonlinear SchrSdinger equation. With these schemes, the solution of the original problem is reduced to the solution of the same problem on a much coarser grid together with the solutions of two linear problems on a fine grid. We have shown, both theoretically and numerically, that our schemes are efficient and achieve asymptotically optimal accuracy.

A DECOUPLING TWO-GRID METHOD FOR THE STEADY-STATE POISSON-NERNST-PLANCK EQUATIONS

Poisson-Nernst-Planck equations are widely used to describe the electrodiffusion of ions in a solvated biomolecular system. Two kinds of two-grid finite element algorithms are proposed to decouple the steady-state Poisson-Nernst-Planck equations by coarse grid finite element approximations. Both theoretical analysis and numerical experiments show the efficiency and effectiveness of the two-grid algorithms for solving Poisson-Nernst-Planck equations.

Two-Grid Immersed Finite Volume Element Methods for Semi-Linear Elliptic Interface Problems with Non-Homogeneous Jump Conditions

In this paper,we propose an immersed finite volume element method for solving the semi-linear elliptic interface problems with non-homogeneous jump conditions.Furthermore,two-grid techniques are used to improve the computational efficiency.In this way,we only need to solve a non-linear system on the coarse grid,and a linear system on the fine grid.Numerical results illustrate that the proposed method can solve the semi-linear elliptic interface problems efficiently.Approximate secondorder accuracy for the solution in the L¥norm can be obtained for the considered examples.

不对称故障下低电压穿越的多目标解耦控制策略

在传统不对称故障低电压穿越控制中,由于控制自由度有限,并网逆变器控制存在无法同时实现输出电流负序分量和直流侧电压二倍频波动抑制的问题。对此,文中提出一种不对称故障下两级式光伏并网系统低电压穿越的多目标解耦控制策略。该策略将逆变器的控制目标设置为输出电流负序分量抑制,给出了综合考虑逆变器输出电流限幅和无功输出需求的逆变器电流内环控制参考值计算方法;通过双向Buck-Boost变换器将超级电容接入直流母线电容两端维持其电压稳定,并将直流侧电压二倍频波动转移至超级电容输入侧进行抑制。仿真结果表明,相比传统控制方法,所提控制策略有效降低了逆变器三相间的不平衡度,改善了输出电流畸变,减小了直流侧电压二倍频波动。