• Title/Summary/Keyword: superconvergence

Search Result 28, Processing Time 0.02 seconds

SUPERCONVERGENCE OF CRANK-NICOLSON MIXED FINITE ELEMENT SOLUTION OF PARABOLIC PROBLEMS

  • Kwon, Dae Sung;Park, Eun-Jae
    • Korean Journal of Mathematics
    • /
    • v.13 no.2
    • /
    • pp.139-148
    • /
    • 2005
  • In this paper we extend the mixed finite element method and its $L_2$-error estimate for postprocessed solutions by using Crank-Nicolson time-discretization method. Global $O(h^2+({\Delta}t)^2)$-superconvergence for the lowest order Raviart-Thomas element ($Q_0-Q_{1,0}{\times}Q_{0,1}$) are obtained. Numerical examples are presented to confirm superconvergence phenomena.

  • PDF

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
    • /
    • v.4 no.1
    • /
    • pp.67-77
    • /
    • 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.

  • PDF

TWO ORDER SUPERCONVERGENCE OF FINITE ELEMENT METHODS FOR SOBOLEV EQUATIONS

  • Li, Qian;Wei, Hong
    • Journal of applied mathematics & informatics
    • /
    • v.8 no.3
    • /
    • pp.721-729
    • /
    • 2001
  • We consider finite element methods applied to a class of Sobolev equations in $R^d$($d{\geq}1$). Global strong superconvergence, which only requires that partitions are quais-uniform, is investigated for the error between the approximate solution and the Ritz-Sobolev projection of the exact solution. Two order superconvervgence results are demonstrated in $W^{1,p}({\Omega})$ and $L_p({\Omega})$ for $2{\leq}p$${\infty}$.

SUPERCONVERGENCE AND POSTPROCESSING OF EQUILIBRATED FLUXES FOR QUADRATIC FINITE ELEMENTS

  • KWANG-YEON KIM
    • Journal of the Korean Society for Industrial and Applied Mathematics
    • /
    • v.27 no.4
    • /
    • pp.245-271
    • /
    • 2023
  • In this paper we discuss some recovery of H(div)-conforming flux approximations from the equilibrated fluxes of Ainsworth and Oden for quadratic finite element methods of second-order elliptic problems. Combined with the hypercircle method of Prager and Synge, these flux approximations lead to a posteriori error estimators which provide guaranteed upper bounds on the numerical error. Furthermore, we prove some superconvergence results for the flux approximations and asymptotic exactness for the error estimator under proper conditions on the triangulation and the exact solution. The results extend those of the previous paper for linear finite element methods.

SUPERCONVERGENCE AND A POSTERIORI ERROR ESTIMATES OF VARIATIONAL DISCRETIZATION FOR ELLIPTIC CONTROL PROBLEMS

  • Hua, Yuchun;Tang, Yuelong
    • Journal of applied mathematics & informatics
    • /
    • v.32 no.5_6
    • /
    • pp.707-719
    • /
    • 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.

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
    • /
    • v.31 no.3_4
    • /
    • pp.479-490
    • /
    • 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.

SUPERCONVERGENCE OF FINITE ELEMENT METHODS FOR LINEAR QUASI-PARABOLIC INTEGRO-DIFFERENTIAL EQUATIONS

  • Li, Qian;Shen, Wanfang;Jian, Jinfeng
    • Journal of the Korean Society for Industrial and Applied Mathematics
    • /
    • v.8 no.2
    • /
    • pp.23-38
    • /
    • 2004
  • We consider finite element methods applied to a class of quasi parabolic integro-differential equations in $R^d$. Global strong superconvergence, which only requires that partitions are quasi-uniform, is investigated for the error between the approximate solution and the Sobolev-Volterra projection of the exact solution. Two order superconvergence results are demonstrated in $W^{1,p}(\Omega)\;and\;L_p(\Omega)$, for $2\;{\leq}p\;<\;{\infty}$.

  • PDF

[ $L_p$ ] ERROR ESTIMATES AND SUPERCONVERGENCE FOR FINITE ELEMENT APPROXIMATIONS FOR NONLINEAR HYPERBOLIC INTEGRO-DIFFERENTIAL PROBLEMS

  • Li, Qian;Jian, Jinfeng;Shen, Wanfang
    • Journal of the Korean Society for Industrial and Applied Mathematics
    • /
    • v.9 no.1
    • /
    • pp.17-29
    • /
    • 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.

  • PDF

SUPERCONVERGENCE OF HYBRIDIZABLE DISCONTINUOUS GALERKIN METHOD FOR SECOND-ORDER ELLIPTIC EQUATIONS

  • MOON, MINAM;LIM, YANG HWAN
    • Journal of the Korean Society for Industrial and Applied Mathematics
    • /
    • v.20 no.4
    • /
    • pp.295-308
    • /
    • 2016
  • We propose a projection-based analysis of a new hybridizable discontinuous Gale-rkin method for second order elliptic equations. The method is more advantageous than the standard HDG method in a sense that the new method has higher-order accuracy and lower computational cost, and is more flexible. Notable distinctions of our new method, when compared to the standard HDG emthod, are that our method uses $L^2$-projection and suitable stabilization parameter depending on a mesh size for superconvergence. We show that the error for the solution of the equation converges with order p + 2 when we only use polynomials of degree p + 1 as a finite element space without postprocessing. After establishing the theory, we carry out numerical tests to demonstrate and ensure that the proposed method is effective and accurate in practice.

GENERALIZED DIFFERENCE METHODS FOR ONE-DIMENSIONAL VISCOELASTIC PROBLEMS

  • Li, Huanrong
    • Journal of the Korean Society for Industrial and Applied Mathematics
    • /
    • v.9 no.2
    • /
    • pp.55-64
    • /
    • 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.

  • PDF