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15 September 2012, Volume 30 Issue 5
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THE BEST
L
^{2}
NORM ERROR ESTIMATE OF LOWER ORDERFINITE ELEMENT METHODS FOR THE FOURTH ORDERPROBLEM
Jun Hu, ZhongCi Shi
2012, 30(5): 449460. DOI:
10.4208/jcm.1203m3855
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In the paper, we analyze the
L
^{2}
norm error estimate of lower order finite elementmethods for the fourth order problem. We prove that the best error estimate in the
L
^{2}
norm of the finite element solution is of second order, which can not be improved generally.The main ingredients are the saturation condition established for these elements and anidentity for the error in the energy norm of the finite element solution. The result holdsfor most of the popular lower order finite element methods in the literature including: thePowellSabin
C
^{1}

P
_{2}
macro element, the nonconforming Morley element, the
C
^{1}

Q
_{2}
macroelement, the nonconforming rectangle Morley element, and the nonconforming incompletebiquadratic element. In addition, the result actually applies to the nonconforming Adinielement, the nonconforming Fraeijs de Veubeke elements, and the nonconforming WangXu element and the WangShiXu element provided that the saturation condition holdsfor them. This result solves one long standing problem in the literature: can the
L
^{2}
normerror estimate of lower order finite element methods of the fourth order problem be twoorder higher than the error estimate in the energy norm?
PRECONDITIONING THE INCOMPRESSIBLE NAVIERSTOKESEQUATIONS WITH VARIABLE VISCOSITY
Xin He, Maya Neytcheva
2012, 30(5): 461482. DOI:
10.4208/jcm.1201m3848
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This paper deals with preconditioners for the iterative solution of the discrete Oseenproblem with variable viscosity. The motivation of this work originates from numericalsimulations of multiphase flow, governed by the coupled CahnHilliard and incompressibleNavierStokes equations. The impact of variable viscosity on some known preconditioningtechnique is analyzed. Theoretical considerations and numerical experiments showthat some broadly used preconditioning techniques for the Oseen problem with constantviscosity are also effcient when the viscosity is varying.
ERROR REDUCTION, CONVERGENCE AND OPTIMALITYFOR ADAPTIVE MIXED FINITE ELEMENT METHODS FORDIFFUSION EQUATIONS
Shaohong Du, Xiaoping Xie
2012, 30(5): 483503. DOI:
10.4208/jcm.1112m3480
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426
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Error reduction, convergence and optimality are analyzed for adaptive mixed finiteelement methods (AMFEM) for diffusion equations without marking the oscillation ofdata. Firstly, the quasierror, i.e. the sum of the stress variable error and the scaled errorestimator, is shown to reduce with a fixed factor between two successive adaptive loops,up to an oscillation. Secondly, the convergence of AMFEM is obtained with respect to thequasierror plus the divergence of the flux error. Finally, the quasioptimal convergencerate is established for the total error, i.e. the stress variable error plus the data oscillation.
A NUMERICAL METHOD FOR SOLVING THE ELLIPTICINTERFACE PROBLEMS WITH MULTIDOMAINS ANDTRIPLE JUNCTION POINTS
Songming Hou, Liqun Wang, Wei Wang
2012, 30(5): 504516. DOI:
10.4208/jcm.1203m3725
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Elliptic interface problems with multidomains and triple junction points have wideapplications in engineering and science. However, the corner singularity makes it a challengingproblem for most existing methods. An accurate and efficient method is desired. Inthis paper, an efficient nontraditional finite element method with nonbodyfitting grids isproposed to solve the elliptic interface problems with multidomains and triple junctions.The resulting linear system of equations is positive definite if the matrix coefficients forthe elliptic equations in the domains are positive definite. Numerical experiments showthat this method is about second order accurate in the
L
^{∞}
norm for piecewise smoothsolutions. Corner singularity can be handled in a way such that the accuracy does notdegenerate. The triple junction is carefully resolved and it does not need to be placedon the grid, giving our method the potential to treat moving interface problems withoutregenerating mesh.
STABILITY AND RESONANCES OF MULTISTEP COSINEMETHODS
B. Cano, M.J. Moreta
2012, 30(5): 517532. DOI:
10.4208/jcm.1203m3487
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383
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In a previous paper, some particular multistep cosine methods were constructed whichproved to be very efficient because of being able to integrate in a stable and explicit waylinearly stiff problems of secondorder in time. In the present paper, the conditions whichguarantee stability for general methods of this type are given, as well as a thorough studyof resonances and filtering for symmetric ones (which, in another paper, have been provedto behave very advantageously with respect to conservation of invariants in Hamiltonianwave equations). What is given here is a systematic way to analyse and treat any of themethods of this type in the mentioned aspects.
BANDED TOEPLITZ PRECONDITIONERS FOR TOEPLITZMATRICES FROM SINC METHODS
ZhiRu Ren
2012, 30(5): 533543. DOI:
10.4208/jcm.1203m3761
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We give general expressions, analyze algebraic properties and derive eigenvalue boundsfor a sequence of Toeplitz matrices associated with the sinc discretizations of various ordersof differential operators. We demonstrate that these Toeplitz matrices can be satisfactorilypreconditioned by certain banded Toeplitz matrices through showing that the spectra ofthe preconditioned matrices are uniformly bounded. In particular, we also derive eigenvaluebounds for the banded Toeplitz preconditioners. These results are elementary inconstructing highquality structured preconditioners for the systems of linear equationsarising from the sinc discretizations of ordinary and partial differential equations, and areuseful in analyzing algebraic properties and deriving eigenvalue bounds for the correspondingpreconditioned matrices. Numerical examples are given to show effectiveness of thebanded Toeplitz preconditioners.
A PVERSION TWO LEVEL SPLINE METHOD FORSEMILINEAR ELLIPTIC EQUATIONS
Xinping Shao, Danfu Han, Xianliang Hu
2012, 30(5): 544554. DOI:
10.4208/jcm.1203m3813
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462
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A novel two level spline method is proposed for semilinear elliptic equations, wherethe two level iteration is implemented between a pair of hierarchical spline spaces withdifferent orders. The new two level method is implementation in a manner of padaptivity.A coarse solution is obtained from solving the model problem in the low order spline space,and the solution with higher accuracy are generated subsequently, via one step Newtonor monidifed Newton iteration in the high order spline space. We also derive the optimalerror estimations for the proposed two level schemes. At last, the illustrated numericalresults confirm our error estimations and further research topics are commented.
THE ULTRACONVERGENCE OF EIGENVALUES FORBIQUADRATIC FINITE ELEMENTS
Lingxiong Meng, Zhimin Zhang
2012, 30(5): 555564. DOI:
10.4208/jcm.1203m3977
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416
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The classical eigenvalue problem of the secondorder elliptic operator is approximatedwith biquadratic nite element in this paper. We construct a new superconvergent functionrecovery operator, from which the
O
(
h
^{8}
ln
h

^{2}
) ultraconvergence of eigenvalue approximation is obtained. Numerical experiments verify the theoretical results.
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