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A Legendre-collocation method is proposed to solve the nonlinear Volterra integral equations of the second kind. We provide a rigorous error analysis for the proposed method, which indicate that the numerical errors in L2-norm and L∞-norm will decay exponentially provided that the kernel function is sufficiently smooth. Numerical results are presented, which confirm the theoretical prediction of the exponential rate of convergence.
A two-grid method for solving the Cahn-Hilliard equation is proposed in this paper. This two-grid method consists of two steps. First, solve the Cahn-Hilliard equation with an implicit mixed finite element method on a coarse grid. Second, solve two Poisson equations using multigrid methods on a fine grid. This two-grid method can also be combined with local mesh refinement to further improve the efficiency. Numerical results including two and three dimensional cases with linear or quadratic elements show that this two-grid method can speed up the existing mixed finite method while keeping the same convergence rate.
be a graph of order
with minimum degree
. Arkin and Hassin [‘Graph partitions with minimum degree constraints’, Discrete Math.190 (1998), 55–65] conjectured that there exists a bipartition
$\lfloor n/2\rfloor -2\leq |S|,|T|\leq \lceil n/2\rceil +2$
and the minimum degrees in the subgraphs induced by
are at least two. In this paper, we first show that
has a bipartition such that the minimum degree in each part is at least two, and then prove that the conjecture is true if the complement of
contains no complete bipartite graph
$r=\lfloor n/2\rfloor -3$
In this paper, the Crank-Nicolson linear finite volume element method is applied to solve the distributed optimal control problems governed by a parabolic equation. The optimal convergent order O(h2+k2) is obtained for the numerical solution in a discrete L2-norm. A numerical experiment is presented to test the theoretical result.
In this paper, we discuss the a posteriori error estimates of the mixed finite element method for quadratic optimal control problems governed by linear parabolic equations. The state and the co-state are discretized by the high order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a posteriori error estimates for both the state and the control approximation. Such estimates, which are apparently not available in the literature, are an important step towards developing reliable adaptive mixed finite element approximation schemes for the control problem.
Two different short period superlattice (SPS) structures with nominally equivalent lattice mismatch, InAs/AlAs and InAs/GaAs are examined using in situ scanning tunneling microscopy (STM). Depending upon the growth conditions, the composition of the InAs/AlAs SPS structure can be either homogeneous or modulated in the lateral direction. Distinct periodic structures are clearly visible in images of the modulated SPS while no periodicity is observed in the homogeneous SPS. For a 30 period SPS consisting of 2 monolayers of AlAs and 2 monolayers of InAs we observe structures 20 nm in size with an average spacing of ∼25 nm in orthogonal directions, which is approximately the same length scale as the composition modulation. Despite the nominally equivalent lattice mismatch, the InAs/GaAs SPS structures are quite different. Some degree of modulation is always observed in these structures. Homogeneous structures are not observed. For the modulated SPS structures, scanning tunneling spectroscopy can be used to characterize the chemical profile.
A glass-ceramic (GC0) with nominal composition of 51.2% CaO–12.1% MgO–36.7% SiO2 (wt%) was synthesized. Then multiphase glass-ceramics of MGC1 and MGC2 were obtained by adding 1 and 2 wt% B2O3 to GC0 followed by thermal treatment. The bending strength of MGC1 was the highest, about 89.46 MPa, and the coefficient of thermal expansion was 10.67 × 10−6 °C−1, closer to that of Ti–6Al–4V alloy (10.03 × 10−6 °C−1). X-ray diffraction analysis confirmed that MGC1 was predominantly composed of akermanite, merwinite, and small amounts of dicalcium silicate crystalline phases. The bioactivity and cytocompatibility in vitro of MGC1 were detected by investigating the bonelike apatite-formation ability in simulated body fluid (SBF) and osteoblast morphology and viability. The results showed that MGC1 possessed bonelike apatite-formation ability in SBF and could release ionic products to significantly stimulate cell growth and viability. Furthermore, osteoblasts adhered and spread well on MGC1, indicating good bioactivity and potential cytocompatibility.
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