On the justification of the Galerkin method for hyperbolic equations

Zhelezovskii, S. E.

doi:10.1134/s0012266107030111

Cited by 4 publications

(7 citation statements)

References 4 publications

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“…This result is common to use for proving existence of exact weak solutions to hyperbolic equations; however, from the view point of justification of the Galerkin method as a constructive tool for approximate solution, the most interesting issue is the proof of strong convergence of approximate solutions to the exact solution and the search of estimates for the convergence rate. Some energy estimates for the convergence rate of the Mikhlin form of the Galerkin method for second-order abstract hyperbolic equations just under the assumption of existence of an exact weak solution were obtained in [6] (for linear equations) and in [7,8] (for quasilinear equations of a less general form than (0.1)). The commonest version of the Galerkin method is the finite element method.…”

Section: Introductionmentioning

confidence: 99%

On error estimates in the Galerkin method for hyperbolic equations

Zhelezovskii

2005

Sib Math J

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We consider the Cauchy problem in a Hilbert space for a second-order abstract quasilinear hyperbolic equation with variable operator coefficients and nonsmooth (but Bochner integrable) free term. For this problem, we establish an a priori energy error estimate for the semidiscrete Galerkin method with an arbitrary choice of projection subspaces. Also, we establish some results on existence and uniqueness of an exact weak solution. We give an explicit error estimate for the finite element method and the Galerkin method in Mikhlin form.Keywords: second-order hyperbolic equation, the Galerkin method, error estimate, weak solution, existence and uniqueness of a solution, finite element method

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Section: Introductionmentioning

confidence: 99%

On error estimates in the Galerkin method for hyperbolic equations

Zhelezovskii

2005

Sib Math J

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“…(3.1) A similar estimate in the case of a linear initial problem and the purely implicit computation scheme (in time) was obtained in [3]. Unlike [3] here we do not require that the operators A(t) and A 0 form an acute angle for the validity of (3.1).…”

Section: Examplesmentioning

confidence: 50%

“…Unlike [3] here we do not require that the operators A(t) and A 0 form an acute angle for the validity of (3.1).…”

Section: Examplesmentioning

confidence: 97%

“…Some energy error estimates for the projection-difference method for abstract hyperbolic equations were earlier established in [3][4][5][6]. In [3], as in this article, the case of a nonsmooth free term was considered but the Galerkin method in Mikhlin form was used for discretization in space variables. Moreover, [3] unlike this article dealt with a linear equation and only the purely implicit scheme (in time).…”

Section: Introductionmentioning

confidence: 98%

“…In [3], as in this article, the case of a nonsmooth free term was considered but the Galerkin method in Mikhlin form was used for discretization in space variables. Moreover, [3] unlike this article dealt with a linear equation and only the purely implicit scheme (in time). In [4][5][6] the author considered the general case of projection subspaces without any special conditions, but the estimates for the error were established for greater smoothness of the exact solution than in this article.…”

Section: Introductionmentioning

confidence: 99%

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Study of convergence of the projection-difference method for hyperbolic equations

Zhelezovskii

2007

Sib Math J

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We consider the Cauchy problem for an abstract quasilinear hyperbolic equation with variable operator coefficients and a nonsmooth but Bochner integrable free term in a Hilbert space. Under study is the scheme for approximate solution of this problem which is a combination of the Galerkin scheme in space variables and the three-layer difference scheme with time weights. We establish an a priori energy error estimate without any special conditions on the projection subspaces. We give a concrete form of this estimate in the case when discretization in the space variables is carried out by the finite element method (for a partial differential equation) and by the Galerkin method in Mikhlin form.

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The Cauchy problem for an abstract second order ordinary differential equation

Gavrilov¹

2020

Sb. Math.

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We prove the existence and uniqueness of a solution for the Cauchy problem for a linear abstract second order differential equation, obtain its representation, and prove that it is continuously dependent on the time at which the initial conditions are specified. Based on these results, we prove the existence and uniqueness of a solution of the Cauchy problem for a nonlinear abstract second order differential equation. This result is applied to show that the initial-boundary value problem for a nonlinear hyperbolic divergence structure equation has a unique solution. Bibliography: 49 titles.

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On the justification of the Galerkin method for hyperbolic equations

Cited by 4 publications

References 4 publications

On error estimates in the Galerkin method for hyperbolic equations

On error estimates in the Galerkin method for hyperbolic equations

Study of convergence of the projection-difference method for hyperbolic equations

The Cauchy problem for an abstract second order ordinary differential equation

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