• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제3권1호
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• pp.1-15
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• 1999

Gradient recovery techniques for the second order elliptic boundary value problem are well known. In particular, the Midpoint and the Vertex Recovery Operator have been studied by various authors and under suitable assumptions on the regularity of the unknown solution superconvergence property of these recovered gradients have been proved. In this paper we extend these results to the recovered gradient of the finite element approximation to a model initial-boundary value problem, and go on to prove superconvergence result for this recovered gradient in a discrete (in time) error norm.

• Korean Journal of Mathematics
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• 제17권1호
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• pp.67-76
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• 2009

We show the existence of at least two nontrivial solutions for a class of the systems of the nonlinear elliptic equations with Dirichlet boundary condition under some conditions for the nonlinear term. We obtain this result by using the variational linking theory in the critical point theory.

• Journal of applied mathematics & informatics
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• 제34권5_6호
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• pp.383-395
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• 2016

The convergence rate of a numerical procedure based on Schwarz Alternating Method (SAM) for solving elliptic boundary value problems depends on the selection of the interface conditions applied on the interior boundaries of the overlapping subdomains. It has been observed that the Robin condition (mixed interface condition), controlled by a parameter, can optimize SAM's convergence rate. In [8], one formulated the twolayer multi-parameterized SAM and determined the optimal values of the multi-parameters to produce the best convergence rate for one-dimensional elliptic boundary value problems. Two-dimensional implementation was presented in [10]. In this paper, we present an implementation for threedimensional problem.

• Journal of applied mathematics & informatics
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• 제33권1_2호
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• pp.33-44
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• 2015

The convergence rate of a numerical procedure based on Schwarz Alternating Method(SAM) for solving elliptic boundary value problems depends on the selection of the interface conditions applied on the interior boundaries of the overlapping subdomains. It has been observed that the Robin condition (mixed interface condition), controlled by a parameter, can optimize SAM's convergence rate. In [7], one formulated the multi-parameterized SAM and determined the optimal values of the multi-parameters to produce the best convergence rate for one-dimensional elliptic boundary value problems. Two-dimensional implementation was presented in [8]. In this paper, we present an implementation for three-dimensional problem.

• Journal of applied mathematics & informatics
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• 제29권1_2호
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• pp.161-171
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• 2011

The convergence rate of a numerical procedure based on Schwarz Alternating Method(SAM) for solving elliptic boundary value problems depends on the selection of the interface conditions applied on the interior boundaries of the overlapping subdomains. It has been observed that the Robin condition (mixed interface condition), controlled by a parameter, can optimize SAM's convergence rate. In [7], one had formulated the multi-parameterized SAM and determined the optimal values of the multi-parameters to produce the best convergence rate for one-dimensional elliptic boundary value problems. However it was not successful for two-dimensional problem. In this paper, we present a new method which utilizes the one-dimensional result to get the optimal convergence rate for the two-dimensional problem.

• 대한수학회지
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• 제54권1호
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• pp.249-265
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• 2017

Interface problem here refers to a second order elliptic problem with a discontinuous coefficient for the second order derivatives. For the corresponding boundary value problem, the maximum principle still holds but Hopf's boundary point lemma may fail. We will give an optimal power type estimate that replaces Hopf's lemma at those boundary points, where this coefficient jumps.

• Korean Journal of Mathematics
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• 제19권1호
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• pp.101-109
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• 2011

We get a theorem which shows that there exist at least two or three nontrivial weak solutions for the nonlinear parabolic boundary value problem with the variable coefficient jumping nonlinearity. We prove this theorem by restricting ourselves to the real Hilbert space. We obtain this result by approaching the topological method. We use the Leray-Schauder degree theory on the real Hilbert space.

• Korean Journal of Mathematics
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• 제25권3호
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• pp.455-468
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• 2017

We prove the existence of multiple solutions for the fourth order nonlinear elliptic problem with fully nonlinear term. Our method is based on the critical point theory; the variation of linking method and category theory.

• Journal of applied mathematics & informatics
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• 제30권3_4호
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• pp.477-488
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• 2012

The convergence rate of a numerical procedure based on Schwarz Alternating Method(SAM) for solving elliptic boundary value problems depends on the selection of the interface conditions applied on the interior boundaries of the overlapping subdomains. It has been observed that the mixed interface condition, controlled by a parameter, can optimize SAM's convergence rate. In [8], one introduced the two-layer multi-parameterized SAM and determined the optimal values of the multi-parameters to produce the best convergence rate for one-dimensional elliptic boundary value problems. In this paper, we present a method which utilizes the one-dimensional result to get the optimal convergence rate for the two-dimensional problem.

• 충청수학회지
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• 제20권1호
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• pp.65-70
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• 2007

We study the existence of nontrivial solutions of the Gradient type Dirichlet boundary value problem for elliptic systems of the form $-{\Delta}U(x)={\nabla}F(x,U(x)),x{\in}{\Omega}$, where ${\Omega}{\subset}R^N(N{\geq}1)$ is a bounded regular domain and U = (u, v) : ${\Omega}{\rightarrow}R^2$. To study the system we use the liking theorem on product space.