• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.5 no.2
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• pp.25-37
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• 2001

We consider the finite element method applied to nonlinear Sobolev equation with smooth data and demonstrate for arbitrary order ($k{\geq}2$) finite element spaces the optimal rate of convergence in $L_{\infty}\;W^{1,{\infty}}({\Omega})$ and $L_{\infty}(L_{\infty}({\Omega}))$ (quasi-optimal for k = 1). In other words, the nonlinear Sobolev equation can be approximated equally well as its linear counterpart. Furthermore, we also obtain superconvergence results in $L_{\infty}(W^{1,{\infty}}({\Omega}))$ for the difference between the approximate solution and the generalized elliptic projection of the exact solution.

• Journal of the Korean housing association
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• v.2 no.1
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• pp.13-34
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• 1991

The main purpose of this study is to examine housing norm of the middle classes, housing norm and normative housing deficits by independent variables(socio - economic variables, family characteristic variable sand housing characteristic variables).There are two major findings of this study as follows :1. In the housing norm, housing space is 99.Om2, the number of rooms is 3.0, housing structure type is apartment, the maintenance cost is 13 thousand won, and housing tenure is home ownership. And housing qualify is classified into 5 dimensions, and neighborhood environment is classified into 3 dimensions.2. This thesis is to conform Morris et aL.(1984)`s hypotheses that cultural norm is homogeneous in culturally unified society and if it appears heterogeneously, It is the subject`s reporting error of the subjects confusing cultural norm with family norm.

• Journal of applied mathematics & informatics
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• v.9 no.1
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• pp.311-320
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• 2002

For second order linear elliptic problems over smooth domains, it is well known that the rate of convergence of the error in the $L_2$norm is one order higher than that in the $H^1$norm. For polygonal domains with reentrant corners, it has been shown in [15] that this extra order cannot be fully recovered when the h-version is used. We present theoretical and computational examples showing the sharpness of our results.

• Bulletin of the Korean Mathematical Society
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• v.53 no.1
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• pp.247-262
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• 2016

We propose and analyze spectral and pseudo-spectral Jacobi-Galerkin approaches for weakly singular Volterra integral equations (VIEs). We provide a rigorous error analysis for spectral and pseudo-spectral Jacobi-Galerkin methods, which show that the errors of the approximate solution decay exponentially in $L^{\infty}$ norm and weighted $L^2$-norm. The numerical examples are given to illustrate the theoretical results.

• Bulletin of the Korean Mathematical Society
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• v.51 no.3
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• pp.717-737
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• 2014

We study the effect of numerical quadrature in space on semidiscrete and fully discrete piecewise linear finite element methods for parabolic interface problems. Optimal $L^2(L^2)$ and $L^2(H^1)$ error estimates are shown to hold for semidiscrete problem under suitable regularity of the true solution in whole domain. Further, fully discrete scheme based on backward Euler method has also analyzed and optimal $L^2(L^2)$ norm error estimate is established. The error estimates are obtained for fitted finite element discretization based on straight interface triangles.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.26 no.10
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• pp.2067-2072
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• 2002

The closed-loop state and input observer is a pole-placement type observer and estimates unknown state and input variables simultaneously. Pole-placement type observers may have poor performances with respect to modeling error and sensing bias error. The effects of these ill-conditioning factors must be minimized for the robust performance in designing observers. In this paper, the steady-state performance of the closed-loop state and input observer is investigated quantitatively and is represented as the estimation error bounds. The performance indices are selected from these error bounds and are related to the robustness with respect to modeling errors and sensing bias. By considering both transient and steady-state performance, the main performance index is determined as the condition number of the eigenvector matrix based on $L_2$-norm.

• Journal of the Korean Mathematical Society
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• v.58 no.3
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• pp.553-569
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• 2021

In this paper, we present and analyze a fully discrete numerical method for solving the time-fractional diffusion wave equation: ∂^β_tu - div(a∇u) = f, 1 < β < 2. We first construct a difference formula to approximate ∂^β_tu by using an interpolation of derivative type. The truncation error of this formula is of O(△t^2+δ-β)-order if function u(t) ∈ C^2,δ[0, T] where 0 ≤ δ ≤ 1 is the Hölder continuity index. This error order can come up to O(△t^3-β) if u(t) ∈ C³ [0, T]. Then, in combinination with the linear finite volume discretization on spatial domain, we give a fully discrete scheme for the fractional wave equation. We prove that the fully discrete scheme is unconditionally stable and the discrete solution admits the optimal error estimates in the H¹-norm and L₂-norm, respectively. Numerical examples are provided to verify the effectiveness of the proposed numerical method.

• KSCE Journal of Civil and Environmental Engineering Research
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• v.9 no.2
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• pp.63-71
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• 1989

The objective of this study is to enhance the reliability of the computed results by setting up of an improved treatments onto the open boundary condition for tidal motion in finite domain. By the $L^2-norm$ and RMS error tests, it was revealed that Sommerfeld's radiating condition gives better result than a forced boundary condition. In the numerical tests for a long wave in a simplified rectangular bay, it was found that the computational accuracy of the newly improved technique to the Sommerfeld condition, suggested in this study with the 2 dimensional shallow finite element model, could be improved by 30% of RMS error to the existing Sommerfeld condition.

• Journal of the Korean Mathematical Society
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• v.51 no.4
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• pp.665-678
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• 2014

We present a new space-time discontinuous Galerkin (DG) method for solving the time dependent, positive symmetric hyperbolic systems. The main feature of this DG method is that the discrete equations can be solved semi-explicitly, layer by layer, in time direction. For the partition made of triangle or rectangular meshes, we give the stability analysis of this DG method and derive the optimal error estimates in the DG-norm which is stronger than the $L_2$-norm. As application, the wave equation is considered and some numerical experiments are provided to illustrate the validity of this DG method.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.15 no.4
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• pp.703-713
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• 2002

Recently, an improved EFG(Element-Free Galerkin) crack analysis technique, which includes a discontinuous approximation and a singular basis function on the auxiliary supports, was developed. The technique is able to accurately analyze the crack propagation problem without any modification of the analysis model; however, it shows some dependency on the analysis parameters used. In this study, the effect of analysis parameters such as the size of compact support, dilation parameter, the smoothness of shape function around the crack tip, and the number of node using auxiliary supports on the accuracy of solution has been investigated. Through a patch test with a crack, relative L₂ error norm of stresses and the stress intensity factor were computed and compared for various analysis parameters and the results were presented as guidelines for adequate choice of analysis parameters.