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15 November 2015, Volume 33 Issue 6
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INEXACT TWOGRID METHODS FOR EIGENVALUE PROBLEMS
Qun Gu, Weiguo Gao
2015, 33(6): 557575. DOI:
10.4208/jcm.1502m4539
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We discuss the inexact twogrid methods for solving eigenvalue problems, including both partial differential and integral equations. Instead of solving the linear system exactly in both traditional twogrid and accelerated twogrid method, we point out that it is enough to apply an inexact solver to the fine grid problems, which will cut down the computational cost. Different stopping criteria for both methods are developed for keeping the optimality of the resulting solution. Numerical examples are provided to verify our theoretical analyses.
AN ADAPTIVE FAST INTERFACE TRACKING METHOD
Yana Di, Jelena Popovic, Olof Runborg
2015, 33(6): 576586. DOI:
10.4208/jcm.1503m4532
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197
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An adaptive numerical scheme is developed for the propagation of an interface in a velocity field based on the fast interface tracking method proposed in[2]. A multiresolution stategy to represent the interface instead of point values, allows local grid refinement while controlling the approximation error on the interface. For time integration, we use an explicit RungeKutta scheme of secondorder with a multiscale time step, which takes longer time steps for finer spatial scales. The implementation of the algorithm uses a dynamic tree data structure to represent data in the computer memory. We briefly review first the main algorithm, describe the essential data structures, highlight the adaptive scheme, and illustrate the computational efficiency by some numerical examples.
STRONG PREDICTORCORRECTOR APPROXIMATION FOR STOCHASTIC DELAY DIFFERENTIAL EQUATIONS
Yuanling Niu, Chengjian Zhang, Kevin Burrage
2015, 33(6): 587605. DOI:
10.4208/jcm.1507m4505
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190
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This paper presents a strong predictorcorrector method for the numerical solution of stochastic delay differential equations(SDDEs) of Itôtype. The method is proved to be meansquare convergent of order min{1/2,
p
} under the Lipschitz condition and the linear growth condition, where
p
is the exponent of Hölder condition of the initial function. Stability criteria for this type of method are derived. It is shown that for certain choices of the flexible parameter
p
the derived method can have a better stability property than more commonly used numerical methods. That is, for some
p
, the asymptotic MSstability bound of the method will be much larger than that of the EulerMaruyama method. Numerical results are reported confirming convergence properties and comparing stability properties of methods with different parameters
p
. Finally, the vectorised simulation is discussed and it is shown that this implementation is much more efficient.
A POSTERIORI ERROR ANALYSIS OF A FULLYMIXED FINITE ELEMENT METHOD FOR A TWODIMENSIONAL FLUIDSOLID INTERACTION PROBLEM
Carolina Domínguez, Gabriel N. Gatica, Salim Meddahi
2015, 33(6): 606641. DOI:
10.4208/jcm.1509m4492
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222
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In this paper we develop an a posteriori error analysis of a fullymixed finite element method for a fluidsolid interaction problem in 2D. The media are governed by the elastodynamic and acoustic equations in timeharmonic regime, respectively, the transmission conditions are given by the equilibrium of forces and the equality of the corresponding normal displacements, and the fluid is supposed to occupy an annular region surrounding the solid, so that a Robin boundary condition imitating the behavior of the Sommerfeld condition is imposed on its exterior boundary. Dualmixed approaches are applied in both domains, and the governing equations are employed to eliminate the displacement u of the solid and the pressure
p
of the fluid. In addition, since both transmission conditions become essential, they are enforced weakly by means of two suitable Lagrange multipliers. The unknowns of the solid and the fluid are then approximated by a conforming Galerkin scheme defined in terms of PEERS elements in the solid, RaviartThomas of lowest order in the fluid, and continuous piecewise linear functions on the boundary. As the main contribution of this work, we derive a reliable and efficient residualbased a posteriori error estimator for the aforedescribed coupled problem. Some numerical results confirming the properties of the estimator are also reported.
NEW TRIGONOMETRIC BASIS POSSESSING EXPONENTIAL SHAPE PARAMETERS
Yuanpeng Zhu, Xuli Han
2015, 33(6): 642684. DOI:
10.4208/jcm.1509m4414
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243
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Four new trigonometric Bernsteinlike basis functions with two exponential shape parameters are constructed, based on which a class of trigonometric Bézierlike curves, analogous to the cubic Bézier curves, is proposed. The corner cutting algorithm for computing the trigonometric Bézierlike curves is given. Any arc of an ellipse or a parabola can be represented exactly by using the trigonometric Bézierlike curves. The corresponding trigonometric Bernsteinlike operator is presented and the spectral analysis shows that the trigonometric Bézierlike curves are closer to the given control polygon than the cubic Bézier curves. Based on the new proposed trigonometric Bernsteinlike basis, a new class of trigonometric Bsplinelike basis functions with two local exponential shape parameters is constructed. The totally positive property of the trigonometric Bsplinelike basis is proved. For different values of the shape parameters, the associated trigonometric Bsplinelike curves can be
C
^{2}
∩
FC
^{3}
continuous for a nonuniform knot vector, and
C
^{3}
or
C
^{5}
continuous for a uniform knot vector. A new class of trigonometric Bézierlike basis functions over triangular domain is also constructed. A de Casteljautype algorithm for computing the associated trigonometric Bézierlike patch is developed. The conditions for
G1 continuous joining two trigonometric Bézierlike patches over triangular domain are deduced.
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