The convergence of Galerkin approximation schemes for second-order hyperbolic equations with dissipation

Kok, B.J.; Geveci, Tunc

doi:10.1090/s0025-5718-1985-0777270-3

Cited by 7 publications

(1 citation statement)

References 10 publications

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“…For error bounds of differentiated errors such choices also had to be employed in [16,22]. Similarly, the right-hand side F h defined in (2.17) satisfies…”

Section: Finally We Introduce the Operators Lmentioning

confidence: 99%

Maximum norm error bounds for the full discretization of nonautonomous wave equations

Dörich,

Leibold,

Maier

2023

IMA Journal of Numerical Analysis

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In the present paper, we consider a specific class of nonautonomous wave equations on a smooth, bounded domain and their discretization in space by isoparametric finite elements and in time by the implicit Euler method. Building upon the work of Baker and Dougalis (1980, On the ${L}^{\infty }$-convergence of Galerkin approximations for second-order hyperbolic equations. Math. Comp., 34, 401–424), we prove optimal error bounds in the $W^{1,\infty } \times L^\infty $-norm for the semidiscretization in space and the full discretization. The key tool is the gain of integrability coming from the inverse of the discretized differential operator. For this, we have to pay with (discrete) time derivatives on the error in the $H^{1} \times L^2$-norm, which are reduced to estimates of the differentiated initial errors. To confirm our theoretical findings, we also present numerical experiments.

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“…For error bounds of differentiated errors such choices also had to be employed in [16,22]. Similarly, the right-hand side F h defined in (2.17) satisfies…”

Section: Finally We Introduce the Operators Lmentioning

confidence: 99%

Maximum norm error bounds for the full discretization of nonautonomous wave equations

Dörich,

Leibold,

Maier

2023

IMA Journal of Numerical Analysis

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Reconstructing initial data using observers: error analysis of the semi-discrete and fully discrete approximations

Haine

Ramdani

2011

Numer. Math.

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International audienceA new iterative algorithm for solving initial data inverse problems from partial observations has been recently proposed in Ramdani et al. (Automatica 46(10), 1616-1625, 2010). Based on the concept of observers (also called Luenberger observers), this algorithm covers a large class of abstract evolution PDE's. In this paper, we are concerned with the convergence analysis of this algorithm. More precisely, we provide a complete numerical analysis for semi-discrete (in space) and fully discrete approximations derived using finite elements in space and an implicit Euler method in time. The analysis is carried out for abstract Schrödinger and wave conservative systems with bounded observation (locally distributed)

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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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The convergence of Galerkin approximation schemes for second-order hyperbolic equations with dissipation

Cited by 7 publications

References 10 publications

Maximum norm error bounds for the full discretization of nonautonomous wave equations

Maximum norm error bounds for the full discretization of nonautonomous wave equations

Reconstructing initial data using observers: error analysis of the semi-discrete and fully discrete approximations

On error estimates in the Galerkin method for hyperbolic equations

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