• 대한수학회보
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• 제50권1호
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• pp.321-341
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• 2013

In this paper, we discuss the a posteriori error estimates of the semidiscrete mixed finite element methods for quadratic optimal control problems governed by linear hyperbolic equations. The state and the co-state are discretized by the order $k$ Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise polynomials of order $k(k{\geq}0)$. Using mixed elliptic reconstruction method, a posterior $L^{\infty}(L^2)$-error estimates for both the state and the control approximation are derived. 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.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제4권1호
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• pp.67-77
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• 2000

In this paper we consider finite element mathods for nonlinear parabolic problems defined in ${\Omega}{\subset}R^d$ ($d{\leq}4$). A new initial approximation 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 $2{\leq}q{\leq}{\infty}$ are demonstrated as well.

• 대한수학회보
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• 제50권4호
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• pp.1127-1144
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• 2013

In this paper, we consider the error estimates of the numerical solutions of a class of fourth order linear-quadratic elliptic optimal control problems by using mixed finite element methods. The state and co-state are approximated by the order $k$ Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise polynomials of order $k(k{\geq}1)$. $L^2$ and $L^{\infty}$-error estimates are derived for both the control and the state approximations. These results are seemed to be new in the literature of the mixed finite element methods for fourth order elliptic control problems.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제6권2호
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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.

• Journal of applied mathematics & informatics
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• 제8권2호
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• pp.327-343
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• 2001

This is a survey article on finite element a posteriori error estimates with an emphasize on gradient recovery type error estimators. As an example, the error estimator based on the ZZ patch recovery technique will be discussed in some detail.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제7권2호
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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 applied mathematics & informatics
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• 제7권1호
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• pp.175-181
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• 2000

The purpose of this to measure, with explicit constants as small as possible, a priori error bounds for approximation by picewise polynomials. These constants play an important role in the numerical verification method of solutions for obstacle problems by using finite element methods .

• 대한수학회보
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• 제51권3호
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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.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제9권1호
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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.

• 한국정밀공학회지
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• 제15권5호
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• pp.120-127
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• 1998

Differential inequalities occurring in problems of obstacle contact problems are recast into variational inequalities and analyzed by finite element methods. A new a posteriori error estimator, which is essential in adaptive finite element method, is introduced to capture the errors in finite element approximations of these variational inequalities. In order to construct a posteriori error estimates, saddle point problems are introduced using Lagrange parameters and upper bounds are provided. The global upper bound is localized by a special mixed formulation, which leads to upper bounds of the element errors. A numerical experiment is performed on an obstacle contact problem to check the effectivity index both in a local and a global sense.