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.

On exact finite-difference schemes for hyperbolic and elliptic equations

2007 ◽  
Vol 43 (7) ◽  
pp. 1001-1010
Author(s):  
P. P. Matus ◽  
V. A. Irkhin ◽  
M. Łapińska-Chrzczonowicz ◽  
S. V. Lemeshevsky
Keyword(s):  
Finite Difference ◽  
Elliptic Equations ◽  
Difference Schemes ◽  
Finite Difference Schemes

A Simple Total-Lagrangian Finite-Element Formulation for Nonlinear Behavior of Planar Beams

2018 ◽  
Author(s):  
Enrico Babilio ◽  
Stefano Lenci
Keyword(s):  
Finite Element ◽  
Theoretical Model ◽  
Finite Difference ◽  
Shear Angle ◽  
Difference Schemes ◽  
Finite Element Formulation ◽  
Finite Difference Schemes ◽  
Element Formulation ◽  
Beam Model ◽  
Lagrangian Finite Element

The present contribution reports some preliminary results obtained applying a simple finite element formulation, developed for discretizing the partial differential equations of motion of a novel beam model. The theoretical model we are dealing with is geometrically exact, with some peculiarities in comparison with other existing models. In order to study its behavior, some numerical investigations have already been performed through finite difference schemes and other methods and are reported in previous contributions. Those computations have enlightened that the model under analysis turns out to be quite hard to handle numerically, especially in dynamics. Hence, we developed ad hoc the total-lagrangian finite-element formulation we report here. The main differences between the theoretical model and its numerical formulation rely on the fact that in the latter the absolute value of the shear angle is assumed to remain much smaller than unity, and strains are piecewise constant along the beam. The first assumption, which actually simplifies equations, has been taken on the basis of results from previous integrations, mainly through finite difference schemes, which clearly showed that, while other strains can achieve large values in their range of admissibility, shear angle actually remains small. The second assumption led us to define a two-nodes constant-strain finite element, with a fast convergence, in terms of number of elements versus solution accuracy. Although, at the present stage of this ongoing research, we have only early results from finite elements, they appear encouraging and start to shed new light on the behavior of the beam model under analysis.

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 Two-layer finite-difference schemes for the Korteweg-de Vries equation in Euler variables

2020 ◽  
Vol 49 ◽  
pp. 57-69
Author(s):  
Vladimir Ivanovich Mazhukin ◽  
◽  
Aleksandr Viktorovich Shapranov ◽  
Elena Nikolaevna Bykovskaya
Keyword(s):  
Finite Difference ◽  
Vries Equation ◽  
Approximation Error ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
The Family ◽  
Implicit Difference Schemes ◽  
Solution Accuracy ◽  
Modified Equation

A family of weighted two-layer finite-difference schemes is presented. Using the example of the numerical solution of model problems on the propagation of a single soliton and the interaction of two solitons, the high quality of explicit-implicit schemes of the Crank-Nichols type with the parameter σ = 0.5 and the order of approximation O(Δt2 + Δx2) isshown. Completely implicit two-layer difference schemes with the parameter σ = 1 and O (Δt+ Δx2) are characterized by absolute stability with a low solution accuracy due to a highapproximation error. The family of explicitly implicit difference schemes is absolutely unstable if the explicitness parameter σ <0.5 prevails. Analysis of the structure of the approximation error, performed using the modified equation method, confirmed the results of numerical simulation.

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