• 제목/요약/키워드: Second-order elliptic equation

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Oscillation of Certain Second Order Damped Quasilinear Elliptic Equations via the Weighted Averages

  • Xia, Yong;Xu, Zhiting
    • Kyungpook Mathematical Journal
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    • 제47권2호
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    • pp.191-202
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    • 2007
  • By using the weighted averaging techniques, we establish oscillation criteria for the second order damped quasilinear elliptic differential equation $$\sum_{i,j=1}^{N}D_i(a_{ij}(x){\parallel}Dy{\parallel}^{p-2}D_jy)+{\langle}b(x),\;{\parallel}Dy{\parallel}^{p-2}Dy{\rangle}+c(x)f(y)=0,\;p>1$$. The obtained theorems include and improve some existing ones for the undamped halflinear partial differential equation and the semilinear elliptic equation.

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Oscillation of Second Order Nonlinear Elliptic Differential Equations

  • Xu, Zhiting
    • Kyungpook Mathematical Journal
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    • 제46권1호
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    • pp.65-77
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    • 2006
  • By using general means, some oscillation criteria for second order nonlinear elliptic differential equation with damping $$\sum_{i,j=1}^{N}D_i[a_{ij}(x)D_iy]+\sum_{i=1}^{N}b_i(x)D_iy+p(x)f(y)=0$$ are obtained. These criteria are of a high degree of generality and extend the oscillation theorems for second order linear ordinary differential equations due to Kamenev, Philos and Wong.

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A MAXIMUM PRINCIPLE FOR NON-NEGATIVE ZEROTH ORDER COEFFICIENT IN SOME UNBOUNDED DOMAINS

  • Cho, Sungwon
    • Korean Journal of Mathematics
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    • 제26권4호
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    • pp.747-756
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    • 2018
  • We study a maximum principle for a uniformly elliptic second order differential operator in nondivergence form. We allow a bounded positive zeroth order coefficient in a certain type of unbounded domains. The results extend a result by J. Busca in a sense of domains, and we present a simple proof based on local maximum principle by Gilbarg and Trudinger with iterations.

ANNULUS CRITERIA FOR OSCILLATION OF SECOND ORDER DAMPED ELLIPTIC EQUATIONS

  • Xu, Zhiting
    • 대한수학회지
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    • 제47권6호
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    • pp.1183-1196
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    • 2010
  • Some annulus oscillation criteria are established for the second order damped elliptic differential equation $$\sum\limits_{i,j=1}^N D_i[a_{ij}(x)D_jy]+\sum\limits_{i=1}^Nb_i(x)D_iy+C(x,y)=0$$ under quite general assumption that they are based on the information only on a sequence of annuluses of $\Omega(r_0)$ rather than on the whole exterior domain $\Omega(r_0)$. Our results are extensions of those due to Kong for ordinary differential equations. In particular, the results obtained here can be applied to the extreme case such as ${\int}_{\Omega(r0)}c(x)dx=-\infty$.

LEAST-SQUARES SPECTRAL COLLOCATION PARALLEL METHODS FOR PARABOLIC PROBLEMS

  • SEO, JEONG-KWEON;SHIN, BYEONG-CHUN
    • 호남수학학술지
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    • 제37권3호
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    • pp.299-315
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    • 2015
  • In this paper, we study the first-order system least-squares (FOSLS) spectral method for parabolic partial differential equations. There were lots of least-squares approaches to solve elliptic partial differential equations using finite element approximation. Also, some approaches using spectral methods have been studied in recent. In order to solve the parabolic partial differential equations in parallel, we consider a parallel numerical method based on a hybrid method of the frequency-domain method and first-order system least-squares method. First, we transform the parabolic problem in the space-time domain to the elliptic problems in the space-frequency domain. Second, we solve each elliptic problem in parallel for some frequencies using the first-order system least-squares method. And then we take the discrete inverse Fourier transforms in order to obtain the approximate solution in the space-time domain. We will introduce such a hybrid method and then present a numerical experiment.

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
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    • 제20권4호
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    • pp.295-308
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    • 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.

ON GROUND STATE SOLUTIONS FOR SINGULAR QUASILINEAR ELLIPTIC EQUATIONS

  • Yin, Honghui;Yang, Zuodong
    • Journal of applied mathematics & informatics
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    • 제29권3_4호
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    • pp.1011-1016
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    • 2011
  • In this paper, our main purpose is to establish the existence of positive bounded entire solutions of second order quasilinear elliptic equation on $R^N$. we obtained the results under different suitable conditions on the locally H$\"{o}$lder continuous nonlinearity f(x, u), we needn't any mono-tonicity condition about the nonlinearity.

PRECONDITIONED SPECTRAL COLLOCATION METHOD ON CURVED ELEMENT DOMAINS USING THE GORDON-HALL TRANSFORMATION

  • Kim, Sang Dong;Hessari, Peyman;Shin, Byeong-Chun
    • 대한수학회보
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    • 제51권2호
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    • pp.595-612
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    • 2014
  • The spectral collocation method for a second order elliptic boundary value problem on a domain ${\Omega}$ with curved boundaries is studied using the Gordon and Hall transformation which enables us to have a transformed elliptic problem and a square domain S = [0, h] ${\times}$ [0, h], h > 0. The preconditioned system of the spectral collocation approximation based on Legendre-Gauss-Lobatto points by the matrix based on piecewise bilinear finite element discretizations is shown to have the high order accuracy of convergence and the efficiency of the finite element preconditioner.

HYBRIDIZABLE DISCONTINUOUS GALERKIN METHOD FOR ELLIPTIC EQUATIONS WITH NONLINEAR COEFFICIENTS

  • MINAM, MOON
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • 제26권4호
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    • pp.244-262
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    • 2022
  • In this paper, we analyze the hybridizable discontinuous Galerkin (HDG) method for second-order elliptic equations with nonlinear coefficients, which are used in many fields. We present the HDG method that uses a mixed formulation based on numerical trace and flux. Under assumptions on the nonlinear coefficient and H2-regularity for a dual problem, we prove that the discrete systems are well-posed and the numerical solutions have the optimal order of convergence as a mesh parameter. Also, we provide a matrix formulation that can be calculated using an iterative technique for numerical experiments. Finally, we present representative numerical examples in 2D to verify the validity of the proof of Theorem 3.10.

AN EFFICIENT IMPLEMENTATION OF BDM MIXED METHODS FOR SECOND ORDER ELLIPTIC PROBLEMS

  • Kim, J.H.
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • 제7권2호
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    • pp.95-111
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    • 2003
  • BDM mixed methods are obtained for a good approximation of velocity for flow equations. In this paper, we study an implementation issue of solving the algebraic system arising from the BDM mixed finite elements. First we discuss post-processing based on the use of Lagrange multipliers to enforce interelement continuity. Furthermore, we establish an equivalence between given mixed methods and projection finite element methods developed by Chen. Finally, we present the implementation of the first order BDM on rectangular grids and show it is as simple as solving the pressure equation.

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