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Immersed finite element method for eigenvalue problem
Lee, S.,Kwak, D.Y.,Sim, I. Koninklijke Vlaamse Ingenieursvereniging ; Elsevie 2017 Journal of computational and applied mathematics Vol.313 No.-
<P>We consider the approximation of elliptic eigenvalue problem with an interface. The main aim of this paper is to prove the stability and convergence of an immersed finite element method (IFEM) for eigenvalues using Crouzeix Raviart P-1-nonconforming approximation. We show that spectral analysis for the classical eigenvalue problem can be easily applied to our model problem. We analyze the IFEM for elliptic eigenvalue problems with an interface and derive the optimal convergence of eigenvalues. Numerical experiments demonstrate our theoretical results. (C) 2016 Elsevier B.V. All rights reserved.</P>
Ren, Y.,Hu, L.,Sakthivel, R. Koninklijke Vlaamse Ingenieursvereniging ; Elsevie 2011 Journal of computational and applied mathematics Vol.235 No.8
This paper deals with the controllability of a class of impulsive neutral stochastic functional differential inclusions with infinite delay in an abstract space. Sufficient conditions for the controllability are derived with the help of the fixed point theorem for discontinuous multi-valued operators due to Dhage. An example is provided to illustrate the obtained theory.
Feng, X.,Kim, I.,Nam, H.,Sheen, D. Koninklijke Vlaamse Ingenieursvereniging ; Elsevie 2011 Journal of computational and applied mathematics Vol.236 No.5
In this paper, we consider locally stabilized pairs (P<SUB>1</SUB>,P<SUB>1</SUB>)-nonconforming quadrilateral and hexahedral finite element methods for the two- and three-dimensional Stokes equations. The stabilization is obtained by adding to the bilinear form the difference between an exact Gaussian quadrature rule for quadratic polynomials and an exact Gaussian quadrature rule for linear polynomials. Optimal error estimates are derived in the energy norm and the L<SUP>2</SUP>-norm for the velocity and in the L<SUP>2</SUP>-norm for the pressure. In addition, numerical experiments to confirm the theoretical results are presented. From our numerical results, we observe that the proposed stabilized(P<SUB>1</SUB>,P<SUB>1</SUB>)-nonconforming finite element method shows better performance than the standard method.
A subspace of the DSSY nonconforming quadrilateral finite element space for the Stokes equations
Park, C.,Sheen, D.,Shin, B.C. Koninklijke Vlaamse Ingenieursvereniging ; Elsevie 2013 Journal of computational and applied mathematics Vol.239 No.-
In this paper, we propose a subspace of the DSSY nonconforming quadrilateral finite element space. The product of this space together with the piecewise constant space can be used for approximating the velocity and pressure variables, respectively, in solving Stokes problems. More precisely, this space consists of the P<SUB>1</SUB>-nonconforming quadrilateral finite element space augmented by macro bubble functions based on the DSSY nonconforming quadrilateral space under a Hood-Taylor type assumption on meshes. It is shown that the pair satisfies the discrete inf-sup condition, using a boundedness estimate of an interpolation operator based on edge integrals. Numerical results are presented.
Iterative methods for solving nonlinear equations with finitely many roots in an interval
Koninklijke Vlaamse Ingenieursvereniging ; Elsevie 2012 Journal of computational and applied mathematics Vol.236 No.13
In this paper we consider a nonlinear equation f(x)=0 having finitely many roots in a bounded interval. Based on the so-called numerical integration method [B.I. Yun, A non-iterative method for solving non-linear equations, Appl. Math. Comput. 198 (2008) 691-699] without any initial guess, we propose iterative methods to obtain all the roots of the nonlinear equation. In the result, an algorithm to find all of the simple roots and multiple ones as well as the extrema of f(x) is developed. Moreover, criteria for distinguishing zeros and extrema are included in the algorithm. Availability of the proposed method is demonstrated by some numerical examples.
A new approach to characterize the solution set of a pseudoconvex programming problem
Son, T.Q.,Kim, D.S. Koninklijke Vlaamse Ingenieursvereniging ; Elsevie 2014 Journal of computational and applied mathematics Vol.261 No.-
A new approach to characterize the solution set of a nonconvex optimization problem via its dual problem is proposed. Some properties of the Lagrange function associated to the problem are investigated. Then characterizations of the solution set of the problem are established.
Superconvergence of new mixed finite element spaces
Hyon, Y.,Kwak, D.Y. Koninklijke Vlaamse Ingenieursvereniging ; Elsevie 2011 Journal of computational and applied mathematics Vol.235 No.14
In this paper we prove some superconvergence of a new family of mixed finite element spaces of higher order which we introduced in [ETNA, Vol. 37, pp. 189-201, 2010]. Among all the mixed finite element spaces having an optimal order of convergence on quadrilateral grids, this space has the smallest unknowns. However, the scalar variable is only suboptimal in general; thus we have employed a post-processing technique for the scalar variable. As a byproduct, we have obtained a superconvergence on a rectangular grid. The superconvergence of a velocity variable naturally holds and can be shown by a minor modification of existing theory, but that of a scalar variable requires a new technique, especially for k=1. Numerical experiments are provided to support the theory.
A posteriori error estimators for the first-order least-squares finite element method
Ku, J.,Park, E.J. Koninklijke Vlaamse Ingenieursvereniging ; Elsevie 2010 Journal of computational and applied mathematics Vol.235 No.1
In this paper, we propose a posteriori error estimators for certain quantities of interest for a first-order least-squares finite element method. In particular, we propose an a posteriori error estimator for when one is interested in @?σ-σ<SUB>h</SUB>@?<SUB>0</SUB> where σ=-A@?u. Our a posteriori error estimators are obtained by assigning proper weight (in terms of local mesh size h<SUB>T</SUB>) to the terms of the least-squares functional. An a posteriori error analysis yields reliable and efficient estimates based on residuals. Numerical examples are presented to show the effectivity of our error estimators.
Kim, K.Y.,Lee, H.C. Koninklijke Vlaamse Ingenieursvereniging ; Elsevie 2010 Journal of computational and applied mathematics Vol.235 No.1
In this work we derive and analyze a posteriori error estimators for low-order nonconforming finite element methods of the linear elasticity problem on both triangular and quadrilateral meshes, with hanging nodes allowed for local mesh refinement. First, it is shown that equilibrated Neumann data on interelement boundaries are simply given by the local weak residuals of the numerical solution. The first error estimator is then obtained by applying the equilibrated residual method with this set of Neumann data. From this implicit estimator we also derive two explicit error estimators, one of which is similar to the one proposed by Dorfler and Ainsworth (2005) [24] for the Stokes problem. It is established that all these error estimators are reliable and efficient in a robust way with respect to the Lame constants. The main advantage of our error estimators is that they yield guaranteed, i.e., constant-free upper bounds for the energy-like error (up to higher order terms due to data oscillation) when a good estimate for the inf-sup constant is available, which is confirmed by some numerical results.
A Crank@?Nicolson scheme for the Landau@?Lifshitz equation without damping
Jeong, D.,Kim, J. Koninklijke Vlaamse Ingenieursvereniging ; Elsevie 2010 Journal of computational and applied mathematics Vol.234 No.2
An accurate and efficient numerical approach, based on a finite difference method with Crank-Nicolson time stepping, is proposed for the Landau-Lifshitz equation without damping. The phenomenological Landau-Lifshitz equation describes the dynamics of ferromagnetism. The Crank-Nicolson method is very popular in the numerical schemes for parabolic equations since it is second-order accurate in time. Although widely used, the method does not always produce accurate results when it is applied to the Landau-Lifshitz equation. The objective of this article is to enumerate the problems and then to propose an accurate and robust numerical solution algorithm. A discrete scheme and a numerical solution algorithm for the Landau-Lifshitz equation are described. A nonlinear multigrid method is used for handling the nonlinearities of the resulting discrete system of equations at each time step. We show numerically that the proposed scheme has a second-order convergence in space and time.