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Finite volume effects in $B_K$ with improved staggered fermions  [PDF]
Jangho Kim,Chulwoo Jung,Hyung-Jin Kim,Weonjong Lee,Stephen R. Sharpe
Physics , 2011, DOI: 10.1103/PhysRevD.83.117501
Abstract: We extend our recent unquenched ($N_f=2+1$ flavor) calculation of $B_K$ using improved staggered fermions by including in the fits the finite volume shift predicted by one-loop staggered chiral perturbation theory. The net result is to lower the result in the continuum limit by 0.6%. This shift is slightly smaller than our previous estimate of finite volume effects based on a direct comparison between different volumes. To include the finite volume effects in a reasonable time, we found it necessary to calculate them using Graphics Processing Units.
Determination of $B_K$ using improved staggered fermions (III) Finite volume effects  [PDF]
Boram Yoon,Taegil Bae,Hyung-Jin Kim,Jangho Kim,Jongjeong Kim,Kwangwoo Kim,Weonjong Lee,Chulwoo Jung,Stephen R. Sharpe,Jisoo Yeo
Physics , 2009,
Abstract: We study the finite-volume effects in our calculation of $B_K$ using HYP-smeared improved staggered valence fermions. We calculate the predictions of both SU(3) and SU(2) staggered chiral perturbation theory at one-loop order. We compare these to the results of a direct calculation, using MILC coarse lattices with two different volumes: $20^3$ and $28^3$. From the direct calculation, we find that the finite volume effect is $\approx 2%$ for the SU(3) analysis and $\approx 0.9%$ for the SU(2) analysis. We also show how the statistical error depends on the number of measurements made per configuration, and make a first study of autocorrelations.
B_K with dynamical overlap fermions  [PDF]
JLQCD Collaboration,N. Yamada,S. Aoki,H. Fukaya,S. Hashimoto,J. Noaki,T. Kaneko,H. Matsufuru,T. Onogi
Physics , 2007,
Abstract: We report on a calculation of $B_K$ with two-flavor dynamical overlap fermions on a $16^3 \times 32$ lattice at $a\sim 0.12$ fm. The results are compared with the PQChPT prediction of quark mass dependence. The systematic errors due to finite volume effects and fixing topology are discussed.
A posteriori estimates for errors of functionals on finite volume approximations to solutions of elliptic boundary value problems  [PDF]
Lutz Angermann
Mathematics , 2012,
Abstract: This article describes the extension of recent methods for a posteriori error estimation such as dual-weighted residual methods to node-centered finite volume discretizations of second order elliptic boundary value problems including upwind discretizations. It is shown how different sources of errors, in particular modeling errors and discretization errors, can be estimated with respect to a user-defined output functional.
Systematic Uncertainties in $B_K$ with Improved Staggered Fermions  [PDF]
Yong-Chull Jang,Taegil Bae,Hyung-Jin Kim,Jangho Kim,Jongjeong Kim,Kwangwoo Kim,Boram Yoon,Weonjong Lee,Chulwoo Jung,Stephen R. Sharpe
Physics , 2010,
Abstract: We study three sources of error in our calculation of $B_K$ using HYP-smeared staggered fermions on the MILC asqtad lattices. These are (1) dependence on the light sea quark mass; (2) finite volume effects; and (3) the impact of an order of magnitude increase in the number of measurements. Our main results are (1) the dependence on the light sea-quark mass is weaker than expected by naive dimensional analysis, (2) including finite volume effects in SU(2) staggered chiral perturbation theory fits leads to a very small change in $B_K$, of size $\approx 0.1%$, and (3) increasing the statistics on one of the coarse MILC lattices resolves a potential discrepancy with other coarse results.
Finite ma Errors of the Overlap Fermion  [PDF]
S. J. Dong,K. F. Liu
Physics , 2007,
Abstract: In this talk, we shall assess the finite ma errors from the overlap fermion. We shall present results on the speed of light from the dispersion relation and hyperfine splitting between the vector and pseudoscalar mesons as a function to ma to reveal the m\Lambda_{QCD}a^2 and m^2a^2 errors. We conclude from this study that one should be limited to using ma less than 0.5 in order to keep the systematic ma errors below a few percent level.
The form factors in the finite volume  [PDF]
V. E. Korepin,N. A. Slavnov
Physics , 1998, DOI: 10.1142/S0217979299002769
Abstract: The form factors of integrable models in finite volume are studied. We construct the explicite representations for the form factors in terms of determinants.
Heavy meson chiral perturbation theory in finite volume  [PDF]
C. -J. David Lin
Physics , 2004, DOI: 10.1016/j.nuclphysbps.2004.11.307
Abstract: We present the first step towards the estimation of finite volume effects in heavy-light meson systems using heavy meson chiral perturbation theory. We demonstrate that these effects can be amplified in both light-quark and heavy-quark mass extrapolations (interpolations) in lattice calculations. As an explicit example, we perform a one-loop calculation for the neutral B meson mixing system and show that finite volume effects, which can be comparable with currently quoted errors, are not negligible in both quenched and partially quenched QCD.
Quenched Finite Volume Logarithms  [PDF]
P. H. Damgaard
Physics , 2001, DOI: 10.1016/S0550-3213(01)00269-3
Abstract: Quenched chiral perturbation theory is used to compute the first finite volume correction to the chiral condensate. The correction diverges logarithmically with the four-volume $V$. We point out that with dynamical quarks one can obtain both the chiral condensate and the pion decay constant from the distributions of the lowest Dirac operator eigenvalues.
Is $2k$-Conjecture valid for finite volume methods?  [PDF]
Waixiang Cao,Zhimin Zhang,Qingsong Zou
Mathematics , 2014,
Abstract: This paper is concerned with superconvergence properties of a class of finite volume methods of arbitrary order over rectangular meshes. Our main result is to prove {\it 2k-conjecture}: at each vertex of the underlying rectangular mesh, the bi-$k$ degree finite volume solution approximates the exact solution with an order $ O(h^{2k})$, where $h$ is the mesh size. As byproducts, superconvergence properties for finite volume discretization errors at Lobatto and Gauss points are also obtained. All theoretical findings are confirmed by numerical experiments.
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