A unified error analysis for the numerical solution of nonlinear wave-type equations with application to kinetic boundary conditions

Leibold, Jan

doi:10.5445/ir/1000130222

Cited by 3 publications

(9 citation statements)

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“…As a specific application, we use this theory to prove error bounds for a non-conforming finite element discretization of the wave equation with nonlinear acoustic boundary conditions. This is a generalization of the results in the thesis [20].…”

Section: Introductionsupporting

confidence: 71%

“…It is possible to include nonlinear forcing terms F Ω (x, u) and F Γ (x, δ) at the right-hand side of (1a) and (1b), respectively. This was considered in [20] for the wave equation with kinetic boundary conditions and such terms can be treated similarly for the acoustic boundary conditions. We omit this here for the sake of a clearer presentation.…”

Section: Remark 21mentioning

confidence: 99%

“…As in the previous section, we first introduce the continuous problem and then present and analyze the abstract space discretization. This is a generalization of the linear unified error analysis introduced in [16] and also an extension of the framework considered in the dissertation [20] which does not cover the acoustic boundary conditions with nonlinear coupling from Sect. 2, cf.Remark 4.2 and Sect.…”

Section: Abstract Space Discretizations Of Second-order Evolution Equ...mentioning

confidence: 99%

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A unified error analysis for nonlinear wave-type equations with application to acoustic boundary conditions

Leibold

2022

Numer. Math.

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In this work we present a unified error analysis for abstract space discretizations of wave-type equations with nonlinear quasi-monotone operators. This yields an error bound in terms of discretization and interpolation errors that can be applied to various equations and space discretizations fitting in the abstract setting. We use the unified error analysis to prove novel convergence rates for a non-conforming finite element space discretization of wave equations with nonlinear acoustic boundary conditions and illustrate the error bound by a numerical experiment.

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Section: Introductionsupporting

confidence: 71%

Section: Remark 21mentioning

confidence: 99%

Section: Abstract Space Discretizations Of Second-order Evolution Equ...mentioning

confidence: 99%

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A unified error analysis for nonlinear wave-type equations with application to acoustic boundary conditions

Leibold

2022

Numer. Math.

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“…The assumptions are satisfied for wave equations with various boundary conditions, in particular of dynamical and Dirichlet type, discretized by nonconforming finite elements, cf. [16,25]. More details are provided in Sections 5 and 6.…”

Section: 4mentioning

confidence: 99%

“…The codes to reproduce the experiments are available at https://doi.org/10.5445/IR/1000134973. The finite element discretization was implemented as described in [25,Ch. 6.5.1].…”

Section: Application: Wave Equation With Kinetic Boundary Conditionsmentioning

confidence: 99%

Full discretization error analysis of exponential integrators for semilinear wave equations

Dörich¹,

Leibold²

2022

Math. Comp.

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In this article we prove full discretization error bounds for semilinear second-order evolution equations. We consider exponential integrators in time applied to an abstract nonconforming semi discretization in space. Since the fully discrete schemes involve the spatially discretized semigroup, a crucial point in the error analysis is to eliminate the continuous semigroup in the representation of the exact solution. Hence, we derive a modified variation-ofconstants formula driven by the spatially discretized semigroup which holds up to a discretization error. Our main results provide bounds for the full discretization errors for exponential Adams and explicit exponential Runge-Kutta methods. We show convergence with the stiff order of the corresponding exponential integrator in time, and errors stemming from the spatial discretization.As an application of the abstract theory, we consider an acoustic wave equation with kinetic boundary conditions, for which we also present some numerical experiments to illustrate our results.

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Strong Norm Error Bounds for Quasilinear Wave Equations Under Weak CFL-Type Conditions

Dörich

2024

Found Comput Math

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In the present paper, we consider a class of quasilinear wave equations on a smooth, bounded domain. We discretize it in space with isoparametric finite elements and apply a semi-implicit Euler and midpoint rule as well as the exponential Euler and midpoint rule to obtain four fully discrete schemes. We derive rigorous error bounds of optimal order for the semi-discretization in space and the fully discrete methods in norms which are stronger than the classical $$H^1\times L^2$$ H 1 × L 2 energy norm under weak CFL-type conditions. To confirm our theoretical findings, we also present numerical experiments.

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A unified error analysis for the numerical solution of nonlinear wave-type equations with application to kinetic boundary conditions

Cited by 3 publications

References 0 publications

A unified error analysis for nonlinear wave-type equations with application to acoustic boundary conditions

A unified error analysis for nonlinear wave-type equations with application to acoustic boundary conditions

Full discretization error analysis of exponential integrators for semilinear wave equations

Strong Norm Error Bounds for Quasilinear Wave Equations Under Weak CFL-Type Conditions

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