• Title/Summary/Keyword: Optimal Error Estimates

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A POSTERIORI L(L2)-ERROR ESTIMATES OF SEMIDISCRETE MIXED FINITE ELEMENT METHODS FOR HYPERBOLIC OPTIMAL CONTROL PROBLEMS

  • Hou, Tianliang
    • Bulletin of the Korean Mathematical Society
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    • v.50 no.1
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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.

A PRIORI ERROR ESTIMATES AND SUPERCONVERGENCE PROPERTY OF VARIATIONAL DISCRETIZATION FOR NONLINEAR PARABOLIC OPTIMAL CONTROL PROBLEMS

  • Tang, Yuelong;Hua, Yuchun
    • Journal of applied mathematics & informatics
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    • v.31 no.3_4
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    • pp.479-490
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    • 2013
  • In this paper, we investigate a priori error estimates and superconvergence of varitional discretization for nonlinear parabolic optimal control problems with control constraints. The time discretization is based on the backward Euler method. The state and the adjoint state are approximated by piecewise linear functions and the control is not directly discretized. We derive a priori error estimates for the control and superconvergence between the numerical solution and elliptic projection for the state and the adjoint state and present a numerical example for illustrating our theoretical results.

ERROR ESTIMATES FOR A SEMI-DISCRETE MIXED DISCONTINUOUS GALERKIN METHOD WITH AN INTERIOR PENALTY FOR PARABOLIC PROBLEMS

  • Ohm, Mi Ray;Lee, Hyun Young;Shin, Jun Yong
    • East Asian mathematical journal
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    • v.32 no.1
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    • pp.101-115
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    • 2016
  • In this paper, we consider a semi-discrete mixed discontinuous Galerkin method with an interior penalty to approximate the solution of parabolic problems. We define an auxiliary projection to analyze the error estimate and obtain optimal error estimates in $L^{\infty}(L^2)$ for the primary variable u, optimal error estimates in $L^2(L^2)$ for ut, and suboptimal error estimates in $L^{\infty}(L^2)$ for the flux variable ${\sigma}$.

QUADRATURE BASED FINITE ELEMENT METHODS FOR LINEAR PARABOLIC INTERFACE PROBLEMS

  • Deka, Bhupen;Deka, Ram Charan
    • 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.

ERROR ESTIMATES OF MIXED FINITE ELEMENT APPROXIMATIONS FOR A CLASS OF FOURTH ORDER ELLIPTIC CONTROL PROBLEMS

  • Hou, Tianliang
    • Bulletin of the Korean Mathematical Society
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    • v.50 no.4
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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.

GENERALIZED DIFFERENCE METHODS FOR ONE-DIMENSIONAL VISCOELASTIC PROBLEMS

  • Li, Huanrong
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.9 no.2
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    • pp.55-64
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    • 2005
  • In this paper, generalized difference methods(GDM) for one-dimensional viscoelastic problems are proposed and analyzed. The new initial values are given in the generalized difference scheme, so we obtain optimal error estimates in $L^p$ and $W^{1,p}(2\;{\leq}\;p\;{\leq}\;{\infty})$ as well as some superconvergence estimates in $W^{1,p}(2\;{\leq}\;p\;{\leq}\;{\infty})$ between the GDM solution and the generalized Ritz-Volterra projection of the exact solution.

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L2-ERROR ANALYSIS OF FULLY DISCRETE DISCONTINUOUS GALERKIN APPROXIMATIONS FOR NONLINEAR SOBOLEV EQUATIONS

  • Ohm, Mi-Ray;Lee, Hyun-Young
    • Bulletin of the Korean Mathematical Society
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    • v.48 no.5
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    • pp.897-915
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    • 2011
  • In this paper, we develop a symmetric Galerkin method with interior penalty terms to construct fully discrete approximations of the solution for nonlinear Sobolev equations. To analyze the convergence of discontinuous Galerkin approximations, we introduce an appropriate projection and derive the optimal $L^2$ error estimates.

OPTIMAL L2-ERROR ESTIMATES FOR EXPANDED MIXED FINITE ELEMENT METHODS OF SEMILINEAR SOBOLEV EQUATIONS

  • Ohm, Mi Ray;Lee, Hyun Young;Shin, Jun Yong
    • Journal of the Korean Mathematical Society
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    • v.51 no.3
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    • pp.545-565
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    • 2014
  • In this paper we derive a priori $L^{\infty}(L^2)$ error estimates for expanded mixed finite element formulations of semilinear Sobolev equations. This formulation expands the standard mixed formulation in the sense that three variables, the scalar unknown, the gradient and the flux are explicitly treated. Based on this method we construct finite element semidiscrete approximations and fully discrete approximations of the semilinear Sobolev equations. We prove the existence of semidiscrete approximations of u, $-{\nabla}u$ and $-{\nabla}u-{\nabla}u_t$ and obtain the optimal order error estimates in the $L^{\infty}(L^2)$ norm. And also we construct the fully discrete approximations and analyze the optimal convergence of the approximations in ${\ell}^{\infty}(L^2)$ norm. Finally we also provide the computational results.

Lp error estimates and superconvergence for finite element approximations for nonlinear parabolic problems

  • LI, QIAN;DU, HONGWEI
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.4 no.1
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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.

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ERROR ESTIMATES FOR FULLY DISCRETE MIXED DISCONTINUOUS GALERKIN APPROXIMATIONS FOR PARABOLIC PROBLEMS

  • OHM, MI RAY;LEE, HYUN YOUNG;SHIN, JUN YONG
    • East Asian mathematical journal
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    • v.31 no.5
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    • pp.685-693
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    • 2015
  • In this paper, we introduce fully discrete mixed discontinuous Galerkin approximations for parabolic problems. And we analyze the error estimates in $l^{\infty}(L^2)$ norm for the primary variable and the error estimates in the energy norm for the primary variable and the flux variable.