New homogenization method for diffusion equations

2018 ◽  
Vol 33 (2) ◽  
pp. 85-93 ◽  
Author(s):  
Yuri A. Kuznetsov
Keyword(s):  
Saddle Point ◽  
Diffusion Coefficients ◽  
Diffusion Problem ◽  
Homogenization Method ◽  
Diffusion Equations ◽  
Point System ◽  
Algebraic Problem ◽  
Saddle Point System ◽  
Saddle Point Matrix ◽  
Diffusion Problems

Abstract In this paper, we propose and investigate a new homogenization method for diffusion problems in domains with multiple inclusions with large values of diffusion coefficients. The diffusion problem is approximated by the P1-finite element method on a triangular mesh. The underlying algebraic problem is replaced by a special system with a saddle point matrix. For the solution of the saddle point system we use the typical asymptotic expansion. We prove the error estimates and convergence of the expanded solutions. Numerical results confirm the theoretical conclusions.

Preconditioning for diffusion problem with small size high resolution inclusions

2018 ◽  
Vol 33 (4) ◽  
pp. 243-251
Author(s):  
Yuri A. Kuznetsov
Keyword(s):  
Solution Method ◽  
Diffusion Problem ◽  
Lanczos Method ◽  
Point System ◽  
High Contrast ◽  
Finite Element Discretization ◽  
Element Discretization ◽  
Preconditioning Technique ◽  
Saddle Point System ◽  
Diffusion Problems

Abstract In this paper, we propose and investigate a new preconditioning technique for diffusion problems with multiple small size high contrast inclusions. The inclusions partitioned into two groups. In the first group inclusions the value of diffusion coefficient can be very small, and in the inclusions of the second group it can be very large. The classical P1 finite element discretization is converted in the special algebraic saddle point system. The solution method combines elimination of the DOFs from the first group of inclusions with the Preconditioned Lanczos method with block diagonal preconditioner for the rest of the DOFs. Condition number estimates for the proposed preconditioner are given.

A simple and high accurate finite volume scheme for diffusion equations

2021 ◽  
pp. 1-6
Author(s):  
Zhucui Jing ◽  
Shuhong Song
Keyword(s):  
Diffusion Coefficients ◽  
Numerical Experiments ◽  
Diffusion Problem ◽  
Diffusion Equations ◽  
Finite Volume Scheme ◽  
Fitting Method ◽  
Accurate Treatment ◽  
Point Scheme ◽  
Discontinuous Diffusion ◽  
Diffusion Problems

Most of numerical methods for diffusion equations, refer to vertex unknowns directly or indirectly, and their accuracy is ultimately determined by the approximation to vertex unknowns. Based on the “twin-fitting” method, a simple and high accurate treatment for the vertex unknowns is developed and is applied to the nine-point scheme for diffusion problem. Numerical experiments show that the resulting nine-point scheme is high accurate for diffusion problems with discontinuous diffusion coefficients on distorted meshes.

Quantum Hall effect in a saddle-point system

1998 ◽  
Vol 2 (1-4) ◽  
pp. 523-526
Author(s):  
M.V Budantsev ◽  
Z.D Kvon ◽  
A.G Pogosov ◽  
E.B Olshanetskii ◽  
D.K Maude ◽  
...  
Keyword(s):  
Saddle Point ◽  
Hall Effect ◽  
Quantum Hall Effect ◽  
Quantum Hall ◽  
Point System ◽  
Saddle Point System

An Iterative Two-Grid Method of A Finite Element PML Approximation for the Two Dimensional Maxwell Problem

2012 ◽  
Vol 4 (2) ◽  
pp. 175-189
Author(s):  
Chunmei Liu ◽  
Shi Shu ◽  
Yunqing Huang ◽  
Liuqiang Zhong ◽  
Junxian Wang
Keyword(s):  
Finite Element ◽  
Saddle Point ◽  
Grid Method ◽  
Variational Problems ◽  
Auxiliary System ◽  
Point System ◽  
Saddle Point System ◽  
System 2 ◽  
System 1 ◽  
Coarse Space

AbstractIn this paper, we propose an iterative two-grid method for the edge finite element discretizations (a saddle-point system) of Perfectly Matched Layer(PML) equations to the Maxwell scattering problem in two dimensions. Firstly, we use a fine space to solve a discrete saddle-point system of H(grad) variational problems, denoted by auxiliary system 1. Secondly, we use a coarse space to solve the original saddle-point system. Then, we use a fine space again to solve a discrete H(curl)-elliptic variational problems, denoted by auxiliary system 2. Furthermore, we develop a regularization diagonal block preconditioner for auxiliary system 1 and use H-X preconditioner for auxiliary system 2. Hence we essentially transform the original problem in a fine space to a corresponding (but much smaller) problem on a coarse space, due to the fact that the above two preconditioners are efficient and stable. Compared with some existing iterative methods for solving saddle-point systems, such as PMinres, numerical experiments show the competitive performance of our iterative two-grid method.

scholarly journals Improving the Solution of Least Squares Support Vector Machines with Application to a Blast Furnace System

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Ling Jian ◽  
Shuqian Shen ◽  
Yunquan Song
Keyword(s):  
Blast Furnace ◽  
Support Vector Machines ◽  
Saddle Point ◽  
Least Squares ◽  
Null Space ◽  
Support Vector ◽  
Point System ◽  
Trend Prediction ◽  
Vector Machines ◽  
Saddle Point System

The solution of least squares support vector machines (LS-SVMs) is characterized by a specific linear system, that is, a saddle point system. Approaches for its numerical solutions such as conjugate methods Sykens and Vandewalle (1999) and null space methods Chu et al. (2005) have been proposed. To speed up the solution of LS-SVM, this paper employs the minimal residual (MINRES) method to solve the above saddle point system directly. Theoretical analysis indicates that the MINRES method is more efficient than the conjugate gradient method and the null space method for solving the saddle point system. Experiments on benchmark data sets show that compared with mainstream algorithms for LS-SVM, the proposed approach significantly reduces the training time and keeps comparable accuracy. To heel, the LS-SVM based on MINRES method is used to track a practical problem originated from blast furnace iron-making process: changing trend prediction of silicon content in hot metal. The MINRES method-based LS-SVM can effectively perform feature reduction and model selection simultaneously, so it is a practical tool for the silicon trend prediction task.

scholarly journals Reduced basis approximations of the solutions to spectral fractional diffusion problems

2020 ◽  
Vol 28 (3) ◽  
pp. 147-160
Author(s):  
Andrea Bonito ◽  
Diane Guignard ◽  
Ashley R. Zhang
Keyword(s):  
Computational Cost ◽  
Fractional Power ◽  
Reaction Diffusion ◽  
Fractional Diffusion ◽  
Diffusion Problem ◽  
Quadrature Point ◽  
Reduced Basis ◽  
Fast Evaluation ◽  
Bottle Neck ◽  
Diffusion Problems

AbstractWe consider the numerical approximation of the spectral fractional diffusion problem based on the so called Balakrishnan representation. The latter consists of an improper integral approximated via quadratures. At each quadrature point, a reaction–diffusion problem must be approximated and is the method bottle neck. In this work, we propose to reduce the computational cost using a reduced basis strategy allowing for a fast evaluation of the reaction–diffusion problems. The reduced basis does not depend on the fractional power s for 0 < smin ⩽ s ⩽ smax < 1. It is built offline once for all and used online irrespectively of the fractional power. We analyze the reduced basis strategy and show its exponential convergence. The analytical results are illustrated with insightful numerical experiments.

scholarly journals The alternate direction iterative methods for generalized saddle point systems

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Shuanghua Luo ◽  
Angang Cui ◽  
Cheng-yi Zhang
Keyword(s):  
Saddle Point ◽  
Iterative Methods ◽  
Iterative Schemes ◽  
Numerical Example ◽  
Convergence Results ◽  
Alternate Direction ◽  
Saddle Point Matrix

Abstract The paper studies two splitting forms of generalized saddle point matrix to derive two alternate direction iterative schemes for generalized saddle point systems. Some convergence results are established for these two alternate direction iterative methods. Meanwhile, a numerical example is given to show that the proposed alternate direction iterative methods are much more effective and efficient than the existing one.

ON A RESCALED NONLOCAL DIFFUSION PROBLEM WITH NEUMANN BOUNDARY CONDITIONS

2021 ◽  
Vol 10 (8) ◽  
pp. 3013-3022
Author(s):  
C.A. Gomez ◽  
J.A. Caicedo
Keyword(s):  
Boundary Conditions ◽  
Existence And Uniqueness ◽  
Neumann Boundary ◽  
Diffusion Problem ◽  
Neumann Boundary Conditions ◽  
Nonlocal Diffusion ◽  
Banach Fixed Point Theorem ◽  
Positive Continuous Function ◽  
Normalization Constant ◽  
Diffusion Problems

In this work, we consider the rescaled nonlocal diffusion problem with Neumann Boundary Conditions \[ \begin{cases} u_t^{\epsilon}(x,t)=\displaystyle\frac{1}{\epsilon^2} \int_{\Omega}J_{\epsilon}(x-y)(u^\epsilon(y,t)-u^\epsilon(x,t))dy\\ \qquad \qquad+\displaystyle\frac{1}{\epsilon}\int_{\partial \Omega}G_\epsilon(x-y)g(y,t)dS_y,\\ u^\epsilon(x,0)=u_0(x), \end{cases} \] where $\Omega\subset\mathbb{R}^{N}$ is a bounded, connected and smooth domain, $g$ a positive continuous function, $J_\epsilon(z)=C_1\frac{1}{\epsilon^N}J(\frac{z}{\epsilon}), G_\epsilon(x)=C_1\frac{1}{\epsilon^N}G(\frac{x}{\epsilon}),$ $J$ and $G$ well defined kernels, $C_1$ a normalization constant. The solutions of this model have been used without prove to approximate the solutions of a family of nonlocal diffusion problems to solutions of the respective analogous local problem. We prove existence and uniqueness of the solutions for this problem by using the Banach Fixed Point Theorem. Finally, some conclusions are given.

scholarly journals Instability of Traveling Pulses in Nonlinear Diffusion-Type Problems and Method to Obtain Bottom-Part Spectrum of Shrödinger Equation with Complicated Potential

2021 ◽  
Author(s):  
Michael I. Tribelsky
Keyword(s):  
Ground State ◽  
Nonlinear Diffusion ◽  
Diffusion Equations ◽  
Nonlinear Diffusion Equations ◽  
Dinger Equation ◽  
Bottom Part ◽  
Diffusion Type ◽  
Traveling Pulses ◽  
General Method ◽  
Diffusion Problems

The instability of traveling pulses in nonlinear diffusion problems is inspected on the example of Gunn domains in semiconductors. Mathematically the problem is reduced to the calculation of the "energy" of the ground state in Schr&ouml;dinger equation with a complicated potential. A general method to obtain the bottom-part spectrum of such equations based on the approximation of the potential by square wells is proposed and applied. Possible generalization of the approach to other types of nonlinear diffusion equations is discussed.

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