A type of full multigrid method for non-selfadjoint Steklov eigenvalue problems in inverse scattering

In this paper, a type of full multigrid method is proposed to solve non-selfadjoint Steklov eigenvalue problems. Multigrid iterations for corresponding selfadjoint and positive definite boundary value problems generate proper iterate solutions that are subsequently added to the coarsest finite element space in order to improve approximate eigenpairs on the current mesh. Based on this full multigrid, we propose a new type of adaptive finite element method for non-selfadjoint Steklov eigenvalue problems. We prove that the computational work of these new schemes are almost optimal, the same as solving the corresponding positive definite selfadjoint boundary value problems. In this case, these type of iteration schemes certainly improve the overfull efficiency of solving the non-selfadjoint Steklov eigenvalue problem. Some numerical examples are provided to validate the theoretical results and the efficiency of this proposed scheme.

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A Multigrid Method for Ground State Solution of Bose-Einstein Condensates

Communications in Computational Physics ◽

10.4208/cicp.191114.130715a ◽

2016 ◽

Vol 19 (3) ◽

pp. 648-662 ◽

Cited By ~ 6

Author(s):

Hehu Xie ◽

Manting Xie

Keyword(s):

Finite Element ◽

Boundary Value Problems ◽

Ground State ◽

Multigrid Method ◽

Eigenvalue Problems ◽

Boundary Value ◽

Ground State Solution ◽

State Solution ◽

Linear Boundary ◽

Bose Einstein

AbstractA multigrid method is proposed to compute the ground state solution of Bose-Einstein condensations by the finite element method based on the multilevel correction for eigenvalue problems and the multigrid method for linear boundary value problems. In this scheme, obtaining the optimal approximation for the ground state solution of Bose-Einstein condensates includes a sequence of solutions of the linear boundary value problems by the multigrid method on the multilevel meshes and some solutions of nonlinear eigenvalue problems some very low dimensional finite element space. The total computational work of this scheme can reach almost the same optimal order as solving the corresponding linear boundary value problem. Therefore, this type of multigrid scheme can improve the overall efficiency for the simulation of Bose-Einstein condensations. Some numerical experiments are provided to validate the efficiency of the proposed method.

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H-Adaptive Finite Element Methods for 1D Stationary High Gradient Boundary Value Problems

Mathematical Modelling and Engineering Problems ◽

10.18280/mmep.080617 ◽

2021 ◽

Vol 8 (6) ◽

pp. 967-973

Author(s):

Collins Olusola Akeremale ◽

Oluwasegun Adeyemi Olaiju ◽

Su Hoe Yeak

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Numerical Solution ◽

Boundary Value Problems ◽

Boundary Value ◽

Adaptive Finite Element Method ◽

Adaptive Finite Element ◽

High Gradient ◽

Adaptive Fem ◽

Element Method

This article considered the traditional finite element method (FEM) and adaptive finite element method (FEM) for the numerical solution of the one-dimensional boundary value problems. We established the preference or the superiority of the h-adaptive FEM to traditional FEM in high gradient problems in terms of accuracy and cost of computation. Numerical examples which confirm the performance and adaptability of the h-adaptive method over the traditional finite element method and the high accuracy of the numerical solution are presented. Detailed error analysis of linear elements was also discussed. In conclusion, h-adaptive FEM is recommended for complex systems with high gradient problems.

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A Study on Numerical Methodologies in Solving Fluid Flow and Heat Transfer Problems

IOP Conference Series Earth and Environmental Science ◽

10.1088/1755-1315/850/1/012021 ◽

2021 ◽

Vol 850 (1) ◽

pp. 012021

Author(s):

S. Harish ◽

V. Kishorre Annanth ◽

M. Abinash ◽

K.R. Kannan ◽

Sahil Agarwal ◽

...

Keyword(s):

Heat Transfer ◽

Finite Element ◽

Fluid Flow ◽

Numerical Methods ◽

Boundary Value Problems ◽

Eigenvalue Problems ◽

Boundary Value ◽

Free Surface Flows ◽

Boundary Integral ◽

Transfer Problems

Abstract Numerical methods are described as techniques by which several mathematical problems are formulated, because they may be easily solved with arithmetic operations. These methodologies have a great impact on the current development of finite element theory and other areas. We have given a short study of numerical methodologies applied in fluid flow and heat and mass transfer problems in mechanical engineering which includes finite difference method, Finite element method, Boundary value problems (general), Lattice Boltzmann’s methods, Crank-Nicolsan scheme methods, boundary integral method, Runge-Kutta method, Taylor series method and so on. We have discussed some phenomena taking place in fluids such as surface tension, coning, water scattering, Stokes law, gravity-capillary, and unsteady free-surface flows, swirling, and so on. We have also analyzed boundary value problems on boundary problems, eigenvalue problems and found a numerical way to solve these problems. We have presented different numerical methods applied to different fundamental modeling approaches in heat transfer and the performance of the mechanisms (modes) vary concerning the methods applied. The paper is dedicated to demonstrating how the methods are beneficial in solving real-life heat transfer problems in engineering applications. Results of the parameters like thermal conductivity, energy flux, entropy, temperature, etc. have been compared with the existing methods

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