scholarly journals Finite-difference schemes and iterative methods for multidimensional elliptic equations with mixed derivatives

2019 ◽  
Vol 54 (4) ◽  
pp. 454-459 ◽  
Author(s):  
V. M. Volkov ◽  
A. U. Prakonina
Keyword(s):  
Finite Difference ◽  
Iterative Methods ◽  
Spectral Characteristics ◽  
Discontinuous Coefficients ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Difference Problem ◽  
Approximation Formulas ◽  
Mixed Derivatives ◽  
Incomplete Lu Factorization

Finite difference schemes and iterative methods of solving anisotropic diffusion problems governing multidimensional elliptic PDE with mixed derivatives are considered. By the example of the test problem with discontinuous coefficients, it is shown that the spectral characteristics of the finite difference problem and the efficiency of their preconditioning depend on the mixed derivatives approximation method. On the basis of the comparative numerical analysis, the most adequate approximation formulas for the mixed derivatives providing a maximum convergence rate of the bi-conjugate gradients method with the incomplete LU factorization and the Fourier – Jacobi preconditioners are discovered. It is shown that the monotonicity of the finite difference scheme does not guarantee advantages at their iterative implementation. Moreover, the grid maximum principle is not provided under the conditions of essential anisotropy.

On Finite Difference Schemes for Elliptic Equations with Discontinuous Coefficients

2013 ◽  
Vol 13 (3) ◽  
pp. 281-289
Author(s):  
Manfred Dobrowolski
Keyword(s):  
Finite Element ◽  
Finite Difference ◽  
Elliptic Equations ◽  
Approximation Error ◽  
Discontinuous Coefficients ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
The Finite Element Method ◽  
Order Four ◽  
Element Theory

Abstract. We study the convergence of finite difference schemes for approximating elliptic equations of second order with discontinuous coefficients. Two of these finite difference schemes arise from the discretization by the finite element method using bilinear shape functions. We prove an convergence for the gradient, if the solution is locally in H3. Thus, in contrast to the first order convergence for the gradient obtained by the finite element theory we show that the gradient is superclose. From the Bramble–Hilbert Lemma we derive a higher order compact (HOC) difference scheme that gives an approximation error of order four for the gradient. A numerical example is given.

FINITE DIFFERENCE SCHEMES WITH CROSS DERIVATIVES CORRECTORS FOR MULTIDIMENSIONAL PARABOLIC SYSTEMS

2006 ◽  
Vol 03 (01) ◽  
pp. 27-52 ◽  
Author(s):  
FRANÇOIS BOUCHUT ◽  
HERMANO FRID
Keyword(s):  
Finite Difference ◽  
Parabolic Systems ◽  
Finite Volume Methods ◽  
Difference Schemes ◽  
Navier Stokes ◽  
Finite Difference Schemes ◽  
First Order ◽  
Flux Vector Splitting ◽  
Cross Derivatives ◽  
Mixed Derivatives

We propose finite difference schemes for multidimensional quasilinear parabolic systems whose main feature is the introduction of correctors which control the second-order terms with mixed derivatives. We show that with these correctors the schemes inherit physically relevant properties present at the continuous level, such as the existence of invariant domains and/or the nonincrease of the total amount of entropy. The analysis is performed with some general tools that could be used also in the analysis of finite volume methods based on flux vector splitting for first-order hyperbolic problems on unstructured meshes. Applications to the compressible Navier–Stokes system are given.

scholarly journals Iterative Methods for Obtaining Energy-Minimizing Parametric Snakes with Applications to Medical Imaging

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Alexandru Ioan Mitrea ◽  
Radu Badea ◽  
Delia Mitrea ◽  
Sergiu Nedevschi ◽  
Paulina Mitrea ◽  
...  
Keyword(s):  
Medical Imaging ◽  
Numerical Methods ◽  
Finite Difference ◽  
Iterative Method ◽  
Iterative Methods ◽  
Approximation Error ◽  
Deformable Models ◽  
Difference Schemes ◽  
Finite Difference Schemes

After a brief survey on the parametric deformable models, we develop an iterative method based on the finite difference schemes in order to obtain energy-minimizing snakes. We estimate the approximation error, the residue, and the truncature error related to the corresponding algorithm, then we discuss its convergence, consistency, and stability. Some aspects regarding the prosthetic sugical methods that implement the above numerical methods are also pointed out.

scholarly journals Iterative realization of finite difference schemes in the fictitious domain method for elliptic problems with mixed derivatives

2019 ◽  
pp. 69-76 ◽  
Author(s):  
Vasily M. Volkov ◽  
Alena V. Prakonina
Keyword(s):  
Finite Difference ◽  
Small Parameter ◽  
Iterative Methods ◽  
Difference Schemes ◽  
Fictitious Domain ◽  
Finite Difference Schemes ◽  
Fictitious Domain Method ◽  
Step Size ◽  
Asymptotic Dependence ◽  
Domain Method

Development of efficient finite difference schemes and iterative methods for solving anisotropic diffusion problems in an arbitrary geometry domain is considered. To simplify the formulation of the Neumann boundary conditions, the method of fictitious domains is used. On the example of a two-dimensional model problem of potential distribution in an isolated anisotropic ring conductor a comparative efficiency analysis of some promising finite-difference schemes and iterative methods in terms of their compatibility with the fictitious domain method is carried out. On the basis of numerical experiments empirical estimates of the asymptotic dependence of the convergence rate of the biconjugate gradient method with Fourier – Jacobi and incomplete LU factorization preconditioners on the step size and the value of the small parameter determining the continuation of the conductivity coefficient in the fictitious domain method are obtained. It is shown, that for one of the considered schemes the Fourier – Jacobi preconditioner is spectrally optimal and allows to eliminate the asymptotical dependence of the iterations number to achieve a given accuracy both on the value of the step size and the value of the small parameter in the fictitious domain method.

scholarly journals Optimizing the spectral characteristics of the finite-difference schemes for the unsteady Schrödinger equation

2019 ◽  
Vol 55 (1) ◽  
pp. 62-68 ◽  
Author(s):  
A. N. Hureuski
Keyword(s):  
Finite Difference ◽  
Schrödinger Equation ◽  
Spectral Range ◽  
Schrodinger Equation ◽  
Spectral Characteristics ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Iir Filters ◽  
Order Scheme ◽  
Theta Method

The spectral consistency of the finite-difference theta-method for the unsteady Schrödinger equation is investigated. Optimal sampling parameters providing a minimum error for a given spectral range are obtained. It is shown that the op ti mized scheme provides a reduction (by a factor of 5–6) in the error of the approximate solution in comparison with the 4th order accuracy scheme. It is shown that the 4th order scheme provides the best spectral consistency only in the case if the spectral range length tends to zero. The conditions for equivalence between the finite-difference scheme and the scheme in the form of two first-order conjugated IIR filters are found. The obtained scheme is the best scheme in the class of conservative finite difference schemes for solving the Schrödinger equation. Practical issues arising in the process of implementing a numerical solution are considered. The obtained results can be efficiently used for solving linear and non-linear Schrödinger equations.

Sign in / Sign up

Export Citation Format

Share Document