• 제목/요약/키워드: optimal $L^2$ error estimates

검색결과 44건 처리시간 0.018초

A PRIORI ERROR ESTIMATES OF A DISCONTINUOUS GALERKIN METHOD FOR LINEAR SOBOLEV EQUATIONS

  • Ohm, Mi-Ray;Shin, Jun-Yong;Lee, Hyun-Young
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • 제13권3호
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    • pp.169-180
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    • 2009
  • A discontinuous Galerkin method with interior penalty terms is presented for linear Sobolev equation. On appropriate finite element spaces, we apply a symmetric interior penalty Galerkin method to formulate semidiscrete approximate solutions. To deal with a damping term $\nabla{\cdot}({\nabla}u_t)$ included in Sobolev equations, which is the distinct character compared to parabolic differential equations, we choose special test functions. A priori error estimate for the semidiscrete time scheme is analyzed and an optimal $L^\infty(L^2)$ error estimation is derived.

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ERROR ESTIMATES FOR FULLY DISCRETE DISCONTINUOUS GALERKIN METHOD FOR NONLINEAR PARABOLIC EQUATIONS

  • Ohm, Mi-Ray;Lee, Hyun-Yong;Shin, Jun-Yong
    • Journal of applied mathematics & informatics
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    • 제28권3_4호
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    • pp.953-966
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    • 2010
  • In this paper, we develop discontinuous Galerkin methods with penalty terms, namaly symmetric interior penalty Galerkin methods to solve nonlinear parabolic equations. By introducing an appropriate projection of u onto finite element spaces, we prove the optimal convergence of the fully discrete discontinuous Galerkin approximations in ${\ell}^2(L^2)$ normed space.

ANALYSIS OF VELOCITY-FLUX FIRST-ORDER SYSTEM LEAST-SQUARES PRINCIPLES FOR THE OPTIMAL CONTROL PROBLEMS FOR THE NAVIER-STOKES EQUATIONS

  • Choi, Young-Mi;Lee, Hyung-Chun
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • 제14권2호
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    • pp.125-140
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    • 2010
  • This paper develops a least-squares approach to the solution of the optimal control problem for the Navier-Stokes equations. We recast the optimality system as a first-order system by introducing velocity-flux variables and associated curl and trace equations. We show that a least-squares principle based on $L^2$ norms applied to this system yields optimal discretization error estimates in the $H^1$ norm in each variable.

A FINITE DIFFERENCE/FINITE VOLUME METHOD FOR SOLVING THE FRACTIONAL DIFFUSION WAVE EQUATION

  • Sun, Yinan;Zhang, Tie
    • 대한수학회지
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    • 제58권3호
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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: ∂βtu - div(a∇u) = f, 1 < β < 2. We first construct a difference formula to approximate ∂βtu by using an interpolation of derivative type. The truncation error of this formula is of O(△t2+δ-β)-order if function u(t) ∈ C2,δ[0, T] where 0 ≤ δ ≤ 1 is the Hölder continuity index. This error order can come up to O(△t3-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.

STABLE LOW ORDER NONCONFORMING QUADRILATERAL FINITE ELEMENTS FOR THE STOKES PROBLEM

  • Kim, Young-Deok;Kim, Se-Ki
    • 대한수학회지
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    • 제39권3호
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    • pp.363-376
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    • 2002
  • Stability result is obtained for the approximation of the stationary Stokes problem with nonconforming elements proposed by Douglas et al [1] for the velocity and discontinuous piecewise constants for the pressure on qudrilateral elements. Optimal order $H^1$and $L^2$error estimates are derived.

ESTRPOLATED CRANK-NICOLSON APPROXIMATION FOR A LINEAR STEFAN PROBLEM WITH A FORCING TERM

  • Ahn, Min-Jung;Lee, Hyun-Young
    • Journal of applied mathematics & informatics
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    • 제8권3호
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    • pp.795-809
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    • 2001
  • In this paper, we apply finite element Galerkin method to a single-ohase linear Stefan problem with a forcing term. We apply the extrapolated Crank-Nicolson method to construct the fully discrete approximation and we derive optimal error estimates in the temporal direction in $L^2$, $H^1$ spaces.

ERROR ESTIMATES OF FULLY DISCRETE DISCONTINUOUS GALERKIN APPROXIMATIONS FOR LINEAR SOBOLEV EQUATIONS

  • Ohm, M.R.;Shin, J.Y.;Lee, H.Y.
    • Journal of applied mathematics & informatics
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    • 제27권5_6호
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    • pp.1221-1234
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    • 2009
  • In this paper, we construct fully discrete discontinuous Galerkin approximations to the solution of linear Sobolev equations. We apply a symmetric interior penalty method which has an interior penalty term to compensate the continuity on the edges of interelements. The optimal convergence of the fully discrete discontinuous Galerkin approximations in ${\ell}^{\infty}(L^2)$ norm is proved.

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ANALYSIS AND COMPUTATIONS OF LEAST-SQUARES METHOD FOR OPTIMAL CONTROL PROBLEMS FOR THE STOKES EQUATIONS

  • Choi, Young-Mi;Kim, Sang-Dong;Lee, Hyung-Chun
    • 대한수학회지
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    • 제46권5호
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    • pp.1007-1025
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    • 2009
  • First-order least-squares method of a distributed optimal control problem for the incompressible Stokes equations is considered. An optimality system for the optimal solution are reformulated to the equivalent first-order system by introducing the vorticity and then the least-squares functional corresponding to the system is defined in terms of the sum of the squared $H^{-1}$ and $L^2$ norms of the residual equations of the system. Finite element approximations are studied and optimal error estimates are obtained. Resulting linear system of the optimality system is symmetric and positive definite. The V-cycle multigrid method is applied to the system to test computational efficiency.

ANALYSIS OF FIRST-ORDER SYSTEM LEAST-SQUARES FOR THE OPTIMAL CONTROL PROBLEMS FOR THE NAVIER-STOKES EQUATIONS

  • Choi, Young-Mi;Kim, Sang-Dong;Lee, Hyung-Chun;Shin, Byeong-Chun
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • 제11권4호
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    • pp.55-68
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    • 2007
  • First-order least-squares method of a distributed optimal control problem for the incompressible Navier-Stokes equations is considered. An optimality system for the optimal solution are reformulated to the equivalent first-order system by introducing velocity-flux variables and then the least-squares functional corresponding to the system is defined in terms of the sum of the squared $L^2$ norm of the residual equations of the system. The optimal error estimates for least-squares finite element approximations are obtained.

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