Some Domain Decomposition and Iterative Refinement Algorithms for Elliptic Finite Element Problems

In this contribution, we report on some results recently obtained in joint work with Maksymilian Dryja. We first study an additive variant of Schwarz' alternating algorithm and establish that a fast method of this kind can be devised which is optimal in the sense that the number of conjugate gradient iterations required, to reach a certain tolerance, is independent of the mesh size as well as the number of subregions.     

Method of Nonconforming Mixed Finite Element for Second Order Elliptic Problems

In this paper, the method of non-conforming mixed finite element for second order elliptic problems is discussed and a format of real optimal order for the lowest order error estimate.     

An Iterative Procedure for Domain Decomposition Method of Second Order Elliptic Problem with Mixed Boundary Conditions

1.IntroductionTherehasbeenaconsiderablenumberofrecentdevelopmentsinnon-overlapdo-maindecompositiontechniquesforsecondorderellipticproblems.WereferespeciallytoMaxiniandQuarteroni[3],[4]andthereferencestherein.Oneofmotivationsforincreasinginterestindomaindecompogitionapproachistodealwithdifferellttypeofequationsindifferentpartsofthephysicaldomain,suchasinthemathematicalmodelingofelasticcompositestructures.InthispaPerwestudyaniterativeprocedurefordomaindecompositionmethodofasimplesecondorderell…     

Multiscale Domain Decomposition Methods for Elliptic Problems with High Aspect Ratios

Abstract In this paper we study some nonoverlapping domain decomposition methods for solving a classof elliptic problems arising from composite materials and flows in porous media which contain many spatialscales. Our preconditioner differs from traditional domain decomposition preconditioners by using a coarsesolver which is adaptive to small scale heterogeneous features. While the convergence rate of traditional domaindecomposition algorithms using coarse solvers based on linear or polynomial interpolations may deteriorate inthe presence of rapid small scale oscillations or high aspect ratios, our preconditioner is applicable to multiple-scale problems without restrictive assumptions and seems to have a convergence rate nearly independent ofthe aspect ratio within the substructures. A rigorous convergence analysis based on the Schwarz framework iscarried out, and we demonstrate the efficiency and robustness of the proposed preconditioner through numericalexperiments which include problems with multipl     

Preconditioning Mixed Finite Element Saddle-point Elliptic Problems

We consider saddle-point problems that typically arise from the mixed finite element discretization of second-order elliptic problems. By proper equivalent algebraic operations the considered saddle-point problem is transformed to another saddle-point problem. The resulting problem can then be efficiently preconditioned by a block-diagonal matrix or by a factored block-matrix (the blocks correspond to the velocity and pressure, respectively). Both preconditioners have a block on the main diagonal that corresponds to the bilinear form (δ is a positive parameter) and a second block that is equal to a constant times the identity operator. We derive uniform bounds for the negative and positive eigenvalues of the preconditioned operator. Then any known preconditioner for the above bilinear form can be applied. We also show some numerical experiments that illustrate the convergence properties of the proposed technique.     

On the Equivalence of Transmission Problems in Nonoverlapping Domain Decomposition Methods for Quasilinear PDEs

We consider a general quasilinear model problem of second order in divergence form on a Lipschitz domain, where the latter is divided arbitrarily in finitely many Lipschitz subdomains. Regarding this decomposition, several transmission problems, being equivalent to the model problem in a weak sense, are constructed. Thereby, no regularity assumption on the solution beyond H ¹ is necessary. Furthermore, we do not need additional smoothness conditions on the boundaries of the subdomains and decompositions with crosspoints are admissible.     

区域分裂法求非线性抛物方程的扩张混合元解

针对非线性抛物方程,给出了全离散的扩张混合元格式,利用一个建立在非重叠型区域分裂技巧上的并行迭代法求解了最后的非线性代数方程组,证明了迭代法的收敛性并给出了最优阶的误差估计.     

Stokes问题的区域分解法

本文主要讨论了Ｓｔｏｋｅｓ问题的非重迭型两仓区域性情形的区域分解算法，首先讨论了连续情形，然后将区域分解算法应用到Ｓｔｏｋｅｓ问题的非协调离散情形。     

Non-overlapping Domain Decomposition Method for a Nodal Finite Element Method

A new approach is proposed for constructing nonoverlapping domain decomposition procedures for solving a linear system related to a nodal finite element method. It applies to problems involving either positive semi-definite or complex indefinite local matrices. The main feature of the method is to preserve the continuity requirements on the unknowns and the finite element equations at the nodes shared by more than two subdomains and to suitably augment the local matrices. We prove that the corresponding algorithm can be seen as a converging iterative method for solving the finite element system and that it cannot break down. Each iteration is obtained by solving uncoupled local finite element systems posed in each subdomain and, in contrast to a strict domain decomposition method, is completed by solving a linear system whose unknowns are the degrees of freedom attached to the above special nodes.     

Mixed Finite Element Methods for Fourth Order Elliptic Optimal Control Problems

In this paper, a priori error estimates are derived for the mixed finite element discretization of optimal control problems governed by fourth order elliptic partial differential equations. The state and co-state are discretized by Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. The error estimates derived for the state variable as well as those for the control variable seem to be new. We illustrate with a numerical example to confirm our theoretical results.     

Decoupled Mixed Element Methods for Fourth Order Elliptic Optimal Control Problems with Control Constraints

In this paper, we study the finite element methods for distributed optimal control problems governed by the biharmonic operator. Motivated from reducing the regularity of solution space, we use the decoupled mixed element method which was used to approximate the solution of biharmonic equation to solve the fourth order optimal control problems. Two finite element schemes, i.e., Lagrange conforming element combined with full control discretization and the nonconforming Crouzeix-Raviart element combined with variational control discretization, are used to discretize the decoupled optimal control system. The corresponding a priori error estimates are derived under appropriate norms which are then verified by extensive numerical experiments.     

The Optimal Preconditioning in the Domain Decomposition Method for Wilson Element

1.TheCollstructionofPreconditionerLetfil)eapolygolldolllaillillR',feL'(fl).Consi(lertheholllogeneousDiricllletboulldaryvalueProblenlofPoissonequation,Assllmethat,fordomainfi,thereareacoarsersubdivisionTHwitllIneshsizeHalldananotheroneThwithmeshsizeh,whichisobtainedbyrefiningTH'Thebotllsubdivisionssatisfythequasi-uniformityandtheillversehypothesis.FOragivenelemelltT,Pm(T)dellotesthespaceofallpolynomialswiththedegreenotgreaterthanm,Qm(T)denotesthespaceofallpolynomialswiththedegreecorres…     

Full-Order Convergence of a Mixed Finite Element Method for Fourth-Order Elliptic Equations

By using a special interpolation operator and an elaborate element analysis, in this paper, we improve the classical error estimates to full order for a mixed finite element method for the fourth-order elliptic equations on the rectangular mesh. Therefore we obtain the truly optimal error estimates in view of the interpolation space for the first time.     

A New Hybridized Mixed Weak Galerkin Method for Second-Order Elliptic Problems

In this paper, a new hybridized mixed formulation of weak Galerkin method is studied for a second order elliptic problem. This method is designed by approximate some operators with discontinuous piecewise polynomials in a shape regular finite element partition. Some discrete inequalities are presented on discontinuous spaces and optimal order error estimations are established. Some numerical results are reported to show super convergence and confirm the theory of the mixed weak Galerkin method.     

Analysis of the Fictitious Domain Method with an L 2-Penalty for Elliptic Problems

The L ²-penalty fictitious domain method is based on a reformulation of the original problem in a larger simple-shaped domain by introducing a discontinuous reaction term with a penalty parameter ε > 0. We first derive regularity results and some a priori estimates and then prove several error estimates. We also give several error estimates for discretization problems by the finite element and finite volume methods.     

对流占优问题稳定化的扩展混合有限元方法

讨论了对流占优问题稳定化的扩展混合元数值模拟.把稳定化的思想与扩展混合元方法相结合,既可以高精度逼近未知函数,未知函数的梯度及伴随向量函数,又能保证格式的稳定性.理论分析表明,方法是有效的,具有最优L²逼近精度.     

A Mixed Finite Element Method for Fourth Order Eigenvalue Problems

A new mixed finite element method for approximating eigenpairs of IV order elliptic eigenvalue problems with Dirichlet boundary conditions has been given. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

解二阶椭圆本征值问题的有限元插值校正方法

§1.引言 本文介绍解二阶椭圆本征值问题的有限元插值校正方法.理论分析和数值实验都表明,此方法具有低代价和高精度的特点.这个方法是超收敛和迭代伽略金法的思想相结合的产物。 考虑二阶椭圆本征值问题:     

非自共轭和不定椭圆问题的有限体积元方法的一致收敛性

In this paper, we prove the existence, uniqueness and uniform convergence of the solution of finite volume element method based on the P1 conforming element for non-selfadjoint and indefinite elliptic problems under minimal elliptic regularity assumption.     

A Priori Error Estimates for Least-Squares Mixed Finite Element Approximation of Elliptic Optimal Control Problems

In this paper, a constrained distributed optimal control problem governed by a first-order elliptic system is considered. Least-squares mixed finite element methods, which are not subject to the Ladyzhenkaya-Babuska-Brezzi consistency condition, are used for solving the elliptic system with two unknown state variables. By adopting the Lagrange multiplier approach, continuous and discrete optimality systems including a primal state equation, an adjoint state equation, and a variational inequality for the optimal control are derived, respectively. Both the discrete state equation and discrete adjoint state equation yield a symmetric and positive definite linear algebraic system. Thus, the popular solvers such as preconditioned conjugate gradient (PCG) and algebraic multi-grid (AMG) can be used for rapid solution. Optimal a priori error estimates are obtained, respectively, for the control function in $L^2(Ω)$-norm, for the original state and adjoint state in $H^1(Ω)$-norm, and for the flux state and adjoint flux state in $H$(div; $Ω$)-norm. Finally, we use one numerical example to validate the theoretical findings.