Semidiscrete and single step fully discrete finite element approximations for second order hyperbolic equations with nonsmooth solutions

Bales, Laurence A.

doi:10.1051/m2an/1993270100551

Cited by 11 publications

(1 citation statement)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Important results for constructing and investigating approximate schemes for the Cauchy problem for second-order evolution equations were obtained by the following authors: G. A. Baker [2], G. A. Baker, J. H. Bramble [3], G. A. Baker, V. A. Dougalis, S. M. Serbin [4], L. A. Bales [5], J. Kačur [18], O. Ladyzhenskaya [20], M. Pultar [28], P. E. Sobolevskij, L. M. Chebotarova [34].…”

Section: Estimations For Two-variable Chebyshev Polynomialsmentioning

confidence: 99%

On local convergence of a symmetric semi-discrete scheme for an abstract analogue of the Kirchhoff equation

Rogava

Tsiklauri

2012

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

We consider the Cauchy problem for a second-order nonlinear evolution equation in a Hilbert space. This equation represents the abstract generalization of the Ball integro-differential equation. The general nonlinear case with respect to terms of the equation which include a square of a norm of a gradient is considered. A three-layer semi-discrete scheme is proposed in order to find an approximate solution. In this scheme, the approximation of nonlinear terms that are dependent on the gradient is carried out by using an integral mean. We show that the solution of the nonlinear discrete problem and its corresponding difference analogue of a first-order derivative is uniformly bounded. For the solution of the corresponding linear discrete problem, it is obtained high-order a priori estimates by using two-variable Chebyshev polynomials. Based on these estimates we prove the stability of the nonlinear discrete problem. For smooth solutions, we provide error estimates for the approximate solution. An iteration method is applied in order to find an approximate solution for each temporal step. The convergence of the iteration process is proved.

show abstract

Section: Estimations For Two-variable Chebyshev Polynomialsmentioning

confidence: 99%

On local convergence of a symmetric semi-discrete scheme for an abstract analogue of the Kirchhoff equation

Rogava

Tsiklauri

2012

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

show abstract

Some remarks on post‐processing and negative norm estimates for approximations to non‐smooth solutions of hyperbolic equations

Bales

1993

Commun. Numer. Meth. Engng.

View full text Add to dashboard Cite

SUMMARYSome formulae and an error estimate for post-processing approximations to solutions of hyperbolic partial differential equations are derived. The post-processing is applied to a fully discrete approximation of the discontinuous solution of an initial boundary-value problem for the wave equation in one space dimension to reduce oscillations and construct a much more accurate approximation to the solution.

show abstract

Study of Three-Layer Semi-Discrete Schemes for Second Order Evolution Equations by Chebyshev Polynomials

Galdava

Gulua

Rogava

2020

Applications of Mathematics and Informatics in Natural Sciences and Engineering

View full text Add to dashboard Cite

Semidiscrete and single step fully discrete finite element approximations for second order hyperbolic equations with nonsmooth solutions

Cited by 11 publications

References 7 publications

On local convergence of a symmetric semi-discrete scheme for an abstract analogue of the Kirchhoff equation

On local convergence of a symmetric semi-discrete scheme for an abstract analogue of the Kirchhoff equation

Some remarks on post‐processing and negative norm estimates for approximations to non‐smooth solutions of hyperbolic equations

Study of Three-Layer Semi-Discrete Schemes for Second Order Evolution Equations by Chebyshev Polynomials

Contact Info

Product

Resources

About