• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.4 no.2
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• pp.41-57
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• 2000

In this paper, we consider first order generalized difference scheme for the two-point boundary value problem and one-dimensional second order parabolic type problem. The optimal error estimates in $L_p$ and $W^{1,p}$ ($2{\leq}p{\leq}{\infty}$) as well as some superconvergence estimates in $W^{1,p}$ ($2{\leq}p{\leq}{\infty}$) are obtained. The main results in this paper perfect the theory of GDM.

• Journal of applied mathematics & informatics
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• v.6 no.1
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• pp.15-30
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• 1999

In this paper the second order semi-discrete and full dis-crete generalized difference schemes for one dimensional parabolic equa-tions are constructed and the optimal order $H^1$ , $L^2$ error estimates and superconvergence results in TEX>$H^1$ are obtained. The results in this paper perfect the theory of generalized difference methods.

• East Asian mathematical journal
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• v.26 no.5
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• pp.615-626
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• 2010

In this paper, we analyze discontinuous Galerkin methods with penalty terms, namly symmetric interior penalty Galerkin methods, to solve nonlinear parabolic equations. We construct finite element spaces on which we develop fully discrete approximations using extrapolated Crank-Nicolson method. We adopt an appropriate elliptic-type projection, which leads to optimal ${\ell}^{\infty}$ ($L^2$) error estimates of discontinuous Galerkin approximations in both spatial direction and temporal direction.

• Journal of applied mathematics & informatics
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• v.32 no.5_6
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• pp.707-719
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• 2014

In this paper, we investigate a variational discretization approximation of elliptic optimal control problems with control constraints. The state and the co-state are approximated by piecewise linear functions, while the control is not directly discretized. By using some proper intermediate variables, we derive a second-order convergence in $L^2$-norm and superconvergence between the numerical solution and elliptic projection of the exact solution in $H^1$-norm or the gradient of the exact solution and recovery gradient in $L^2$-norm. Then we construct a posteriori error estimates by using the superconvergence results and do some numerical experiments to confirm our theoretical results.

• Bulletin of the Korean Mathematical Society
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• v.51 no.1
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• pp.139-156
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• 2014

In this paper, we investigate the error estimates of a quadratic elliptic control problem with pointwise control constraints. The state and the co-state variables are approximated by the order k = 1 Raviart-Thomas mixed finite element and the control variable is discretized by piecewise linear but discontinuous functions. Approximations of order $h^{\frac{3}{2}}$ in the $L^2$-norm and order h in the $L^{\infty}$-norm for the control variable are proved.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.7 no.2
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• pp.35-49
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• 2003

In this paper, finite volume element methods for nonlinear parabolic integrodifferential problems are proposed and analyzed. The optimal error estimates in $L^p\;and\;W^{1,p}\;(2\;{\leq}\;p\;{\leq}\;{\infty})$ as well as some superconvergence estimates in $W^{1,p}\;(2\;{\leq}\;p\;{\leq}\;{\infty})$ are obtained. The main results in this paper perfect the theory of FVE methods.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.9 no.1
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• pp.17-29
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• 2005

In this paper we consider finite element methods for nonlinear hyperbolic integro-differential problems defined in ${\Omega}\;{\subset}\;R^d(d\;{\leq}\;4)$. A new initial approximation of $u_t(0)$ is taken. Optimal order error estimates in $L_p$ for $2\;{\leq}\;p\;{\leq}\;{\infty}$ are established for arbitrary order finite element. One order superconvergence in $W^{1,p}$ for $2\;{\leq}\;p\;{\leq}\;{\infty}$ are demonstrated as well.

• Journal of the Korean Mathematical Society
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• v.34 no.3
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• pp.661-672
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• 1997

In this paper we introduce the semidiscrete solution of a single-phase nonlinear Stefan problem We analyze the optimal convergence of the semidiscrete solution in $H^1$ and $H^2$ normed spaces and also we derive the error estimates in $L^2$ normed space.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.6 no.2
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• pp.85-97
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• 2002

In this paper, finite volume element methods for nonlinear parabolic problems are proposed and analyzed. Optimal order error estimates in $W^{1,p}$ and $L_p$ are derived for $2{\leq}p{\leq}{\infty}$. In addition, superconvergence for the error between the approximation solution and the generalized elliptic projection of the exact solution (or and the finite element solution) is also obtained.

• Proceedings of the Korean Institute of Intelligent Systems Conference
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• 1993.06a
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• pp.1313-1316
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• 1993

This paper presents a fuzzy system that estimates the optimal bit allocation matrices for the spatially active subimage classes of adaptive transform image coding in noisy channels. Transform image coding is good for image data compression but it requires a transmission error protection scheme to maintain the performance since the channel noise degrades its performance. The fuzzy system provides a simple way of estimating the bit allocation matrices from the optimal bit map computed by the method of minimizing the mean square error between the transform coefficients of the original and the reconstructed images.