拟线性Sobolev方程的特征有限元格式的交替方向预处理迭代解及其分析

考虑数值求解具有对流项的高维拟线性Ｓｏｂｏｌｅｖ方程,构造了特征有限元格式,提出用交替方向预处理迭代法求特征有限元格式在每一时间步所产生的代数方程组的近似解,整个计算过程仅对一个可方向交替的预处理矩阵求逆一次,大大降低了计算量．证明了迭代解的最佳L^2模误差估计,并给出了算法的拟优工作量估计．     

半导体问题的交替方向有限元方法

该文用交替方向有限元方法求解半导体问题的Energy Trans port (ET)模型。对模型中椭圆型的电子位势方程采用交替方向迭代法，对流占优扩散的电子浓度和空穴浓度方程采用特征交替方向有限元方法，热传导方程利用Patch逼近采用交替方向有限元方法求解。利用微分方程的先验估计理论和技巧，分别得到了椭圆型方程和抛物型方程的最优H+1和L+2误差估计。     

A new LQP alternating direction method for solving variational inequality problems with separable structure

We presented a new logarithmic-quadratic proximal alternating direction scheme for the separable constrained convex programming problem. The predictor is obtained by solving series of related systems of non-linear equations in a parallel wise. The new iterate is obtained by searching the optimal step size along a new descent direction. The new direction is obtained by the linear combination of two descent directions. Global convergence of the proposed method is proved under certain assumptions. We show the O(1 / t) convergence rate for the parallel LQP alternating direction method.     

Alternating Direction Methods for a System of Parabolic Equations in Two Space Dimensions with a Mixed Derivative

An alternating direction (A.D.I.) method, which requires thesolution of two block tridiagonal sets of equations at eachtime step, is suggested for solving a system of parabolic equationswith variable coefficients in two space dimensions with a mixedderivative. The method is shown to be unconditionally stablefor two semi-infinite ranges of an auxiliary parameter subjectto restrictions on the coefficient matrices. Other existingfinite difference schemes are mentioned and numerical resultsare presented.     

Generalized Alternating Direction Implicit Method for Projected Generalized Lyapunov Equations

We generalize an alternating direction implicit method for projected generalized Lyapunov equations. Low rank versions of this method is also presented that can be used to compute a low rank approximation of the solution of Lyapunov equations with symmetric, positive semidefinite right‐hand side. Numerical example is given. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

An alternating direction Galerkin method for elliptic problems with gradient nonlinearity

Several problems in many applications involve the solution of partial differential equations with gradient dependent nonlinearity. The numerical solution of the resulting nonlinear system is rather expensive. We present an alternating direction Galerkin method that allows much faster solution of the nonlinear system. The alternating direction formulation help reduce the problem into a sequence of nonlinear systems that may be solved very efficiently. Theoretical study of the convergence of the method will also be presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

一类单调变分不等式的非精确交替方向法

交替方向法适合于求解大规模问题.该文对于一类变分不等式提出了一种新的交替方向法.在每步迭代计算中,新方法提出了易于计算的子问题,该子问题由强单调的线性变分不等式和良态的非线性方程系统构成.基于子问题的精确求解,该文证明了算法的收敛性.进一步,又提出了一类非精确交替方向法,每步迭代计算只需非精确求解子问题.在一定的非精确条件下,算法的收敛性得以证明.     

解可分离结构变分不等式的一种新的交替方向法

交替方向法是求解可分离结构变分不等式问题的经典方法之一, 它将一个大型的变分不等式问题分解成若干个小规模的变分不等式问题进行迭代求解. 但每步迭代过程中求解的子问题仍然摆脱不了求解变分不等式子问题的瓶颈. 从数值计算上来说, 求解一个变分不等式并不是一件容易的事情.因此, 本文提出一种新的交替方向法, 每步迭代只需要求解一个变分不等式子问题和一个强单调的非线性方程组子问题. 相对变分不等式问题而言, 我们更容易、且有更多的有效算法求解一个非线性方程组问题. 在与经典的交替方向法相同的假设条件下, 我们证明了新算法的全局收敛性. 进一步的数值试验也验证了新算法的有效性.     

High accuracy A.D.I. methods for fourth order parabolic equations with variable coefficients

High accuracy alternating direction implicit (A.D.I.) methods are derived for solving fourth order parabolic equations with variable coefficients in one, two, and three space dimensions. Splittings are discussed and numerical results are presented.     

An inverse‐free ADI algorithm for computing Lagrangian invariant subspaces

The numerical computation of Lagrangian invariant subspaces of large‐scale Hamiltonian matrices is discussed in the context of the solution of Lyapunov equations. A new version of the low‐rank alternating direction implicit method is introduced, which, in order to avoid numerical difficulties with solutions that are of very large norm, uses an inverse‐free representation of the subspace and avoids inverses of ill‐conditioned matrices. It is shown that this prevents large growth of the elements of the solution that may destroy a low‐rank approximation of the solution. A partial error analysis is presented, and the behavior of the method is demonstrated via several numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.     

Alternating direction collocation for irregular regions

Several methods are presented for solving separable elliptic partial differential equations over an irregular region B using alternating direction collocation on a rectangular grid over an embedding rectangle R. The methods are geometric predictor-corrector schemes. At each iterative step, the numerical solution is predicted on R via a full ADI sweep. The forcing term is then updated (corrected) on collocation points interior to B. By this means, the geometry of B and boundary conditions on ∂B are approximated implicitly using rectangular grids on R. The methods are O(h⁴) in the L²(R) norm and are boundary exact, in that the computed solution converges exactly to the given boundary conditions on ∂B for appropriate choices of a pair of acceleration parameters. A number of examples are presented. Proof of convergence is established elsewhere. © 1996 John Wiley & Sons, Inc.     

The new alternating direction implicit difference methods for solving three-dimensional parabolic equations

A new alternating direction implicit (ADI) scheme for solving three-dimensional parabolic equations with nonhomogeneous boundary conditions is presented. The scheme is also extended to high-order compact difference scheme. Both of them have the advantages of unconditional stability and being convenient to compute the boundary values of the intermediates. Besides this, the compact scheme has high-order accuracy and uses less computational time. Numerical examples are presented and the results are very satisfactory.     

Superconvergent biquadratic finite volume element method for two-dimensional Poisson’s equations

In this paper, a kind of biquadratic finite volume element method is presented for two-dimensional Poisson’s equations by restricting the optimal stress points of biquadratic interpolation as the vertices of control volumes. The method can be effectively implemented by alternating direction technique. It is proved that the method has optimal energy norm error estimates. The superconvergence of numerical gradients at optimal stress points is discussed and it is proved that the method has also superconvergence displacement at nodal points by a modified dual argument technique. Finally, a numerical example verifies the theoretical results and illustrates the effectiveness of the method.     

The construction of hopscotch methods for parabolic and elliptic equations in two space dimensions with a mixed derivative

Hopscotch, a fast finite difference technique, is used to solve parabolic and elliptic equations in two space dimensions with a mixed derivative. The method is compared numerically with existing alternating direction implicit (A.D.I.) and locally one dimensional (L.O.D.) methods for simple problems.Douglas and Gunn's A.D.I. method is both simplified and improved by reformulating it as a hopscotch method.     

Convergence of the alternating direction method for the numerical solution of a heat conduction equation with delay

Two-dimensional parabolic equations with delay effects in the time component are considered. An alternating direction scheme is constructed for the numerical solution of these equations. The question on the reduction of a problem with inhomogeneous boundary conditions to a problem with homogeneous boundary conditions is considered. The order of approximation error for the alternating direction scheme, stability, and convergence order are investigated.     

The new alternating direction implicit difference methods for the wave equations

A new second-order alternating direction implicit (ADI) scheme, based on the idea of the operator splitting, is presented for solving two-dimensional wave equations. The scheme is also extended to a high-order compact difference scheme. Both of them have the advantages of unconditional stability, less impact of the perturbing terms on the accuracy, and being convenient to compute the boundary values of the intermediates. Besides this, the compact scheme has high-order accuracy and costs less in computational time. Numerical examples are presented and the results are very satisfactory.     

AN ADI GALERKIN METHOD WITH MOVING FINITE ELEMENT SPACES FOR A CLASS OF SECOND-ORDER HYPERBOLIC EQUATIONS

1　 IntroductionADI Galerkin methods were first formulated for the solution of nonlinear parabolic andlinear second-order hyperbolic problems on rectangular regions by Douglas and Dupont[1 ] .These methods combine alternating-direction method and Galerkin finite element methodtogether.So,they have the advantage of reducing the solution of a multidimensional problemto the solution of sets of independent one-dimensional problems,decreasing the amount ofcalculation,natural parallelism and highe…     

三维热传导型半导体问题的交替方向有限元方法

1　引　　言三维热传导型半导体器件瞬态问题的数学模型由四个非线性偏微分方程描述[1 ,2 ] ,记 Ω为 Ω=[0 ,1 ] 3的边界 ,三维问题-Δψ =α( p -e+ N( x) ) ,　　 ( x,t)∈Ω× [0 ,T] ,( 1 .1 ) e t= . ( De( x) e-μe( x) e ψ) -R( e,p,T) ,　 ( x,t)∈Ω× ( 0 ,T] ,( 1 .2 ) p t= . ( Dp( x) p +μp( x) p ψ) -R( e,p,T) ,　 ( x,t)∈Ω× ( 0 ,T] ,( 1 .3 )ρ( x) T t-ΔT =[( Dp( x) p +μp( x) p ψ) -( De( x) e-μe( x) e ψ) ] . ψ,　　　　　　 ( x,t)∈Ω× ( 0 ,T] . ( 1 .4 )ψ( x,t) =e( x,t) =p( …