Unified error analysis for nonconforming space discretizations of wave-type equations

2018 ◽  
Vol 39 (3) ◽  
pp. 1206-1245 ◽  
Author(s):  
David Hipp ◽  
Marlis Hochbruck ◽  
Christian Stohrer
Keyword(s):  
Wave Equation ◽  
Error Analysis ◽  
Hilbert Spaces ◽  
Error Bounds ◽  
Wave Equations ◽  
Evolution Equations ◽  
Convergence Rates ◽  
Linear Wave ◽  
Element Discretization ◽  
Continuous Space

Abstract This paper provides a unified error analysis for nonconforming space discretizations of linear wave equations in the time domain. We propose a framework that studies wave equations as first-order evolution equations in Hilbert spaces and their space discretizations as differential equations in finite-dimensional Hilbert spaces. A lift operator maps the semidiscrete solution from the approximation space to the continuous space. Our main results are a priori error bounds in terms of interpolation, data and conformity errors of the method. Such error bounds are the key to the systematic derivation of convergence rates for a large class of problems. To show that this approach significantly eases the proof of new convergence rates, we apply it to an isoparametric finite element discretization of the wave equation with acoustic boundary conditions in a smooth domain. Moreover, our results reproduce known convergence rates for already investigated conforming and nonconforming space discretizations in a concise and unified way. The examples discussed in this paper comprise discontinuous Galerkin discretizations of Maxwell’s equations and finite elements with mass lumping for the acoustic wave equation.

scholarly journals Optimal finite element error estimates for an optimal control problem governed by the wave equation with controls of bounded variation

2020 ◽  
Author(s):  
Sebastian Engel ◽  
Boris Vexler ◽  
Philip Trautmann
Keyword(s):  
Optimal Control ◽  
Finite Element ◽  
Wave Equation ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Bounded Variation ◽  
Convergence Rates ◽  
The State ◽  
Linear Wave ◽  
Element Discretization

Abstract This work discusses the finite element discretization of an optimal control problem for the linear wave equation with time-dependent controls of bounded variation. The main focus lies on the convergence analysis of the discretization method. The state equation is discretized by a space-time finite element method. The controls are not discretized. Under suitable assumptions optimal convergence rates for the error in the state and control variable are proven. Based on a conditional gradient method the solution of the semi-discretized optimal control problem is computed. The theoretical convergence rates are confirmed in a numerical example.

scholarly journals Parameter identification for the linear wave equation with Robin boundary condition

2019 ◽  
Vol 27 (1) ◽  
pp. 25-41
Author(s):  
Valeria Bacchelli ◽  
Dario Pierotti ◽  
Stefano Micheletti ◽  
Simona Perotto
Keyword(s):  
Wave Equation ◽  
Mixed Boundary ◽  
Stability Result ◽  
Interval Length ◽  
Left Endpoint ◽  
Linear Wave ◽  
Reconstruction Procedure ◽  
Element Discretization ◽  
Linear Wave Equation ◽  
Bounded Interval

Abstract We consider an initial-boundary value problem for the classical linear wave equation, where mixed boundary conditions of Dirichlet and Neumann/Robin type are enforced at the endpoints of a bounded interval. First, by a careful application of the method of characteristics, we derive a closed-form representation of the solution for an impulsive Dirichlet data at the left endpoint, and valid for either a Neumann or a Robin data at the right endpoint. Then we devise a reconstruction procedure for identifying both the interval length and the Robin parameter. We provide a corresponding stability result and verify numerically its performance moving from a finite element discretization.

scholarly journals Error Analysis of An Explicit Finite Difference Approximation for the Space Fractional Wave Equations

2012 ◽  
Vol 17 (2) ◽  
pp. 245
Author(s):  
N.H. Sweilam ◽  
T.A. Assiri
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Error Analysis ◽  
Wave Equations ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Fractional Wave Equation ◽  
Fractional Wave Equations ◽  
Explicit Finite Difference ◽  
The Stability

In this paper, the space fractional wave equation (SFWE) is numerically studied, where the fractional derivative is defined in the sense of Caputo. An explicit finite difference approximation (EFDA) for SFWE is presented. The stability and the error analysis of the EFDA are discussed. To demonstrate the effectiveness of the approximated method, some test examples are presented.   

scholarly journals Weak Convergence Rates for Spatial Spectral Galerkin Approximations of Semilinear Stochastic Wave Equations with Multiplicative Noise

2021 ◽  
Author(s):  
Ladislas Jacobe de Naurois ◽  
Arnulf Jentzen ◽  
Timo Welti
Keyword(s):  
Wave Equation ◽  
Weak Convergence ◽  
Wave Equations ◽  
Multiplicative Noise ◽  
Convergence Rates ◽  
Rates Of Convergence ◽  
Kolmogorov Equation ◽  
Continuous Version ◽  
Stochastic Wave Equations ◽  
Galerkin Approximations

AbstractStochastic wave equations appear in several models for evolutionary processes subject to random forces, such as the motion of a strand of DNA in a liquid or heat flow around a ring. Semilinear stochastic wave equations can typically not be solved explicitly, but the literature contains a number of results which show that numerical approximation processes converge with suitable rates of convergence to solutions of such equations. In the case of approximation results for strong convergence rates, semilinear stochastic wave equations with both additive or multiplicative noise have been considered in the literature. In contrast, the existing approximation results for weak convergence rates assume that the diffusion coefficient of the considered semilinear stochastic wave equation is constant, that is, it is assumed that the considered wave equation is driven by additive noise, and no approximation results for multiplicative noise are known. The purpose of this work is to close this gap and to establish essentially sharp weak convergence rates for spatial spectral Galerkin approximations of semilinear stochastic wave equations with multiplicative noise. In particular, our weak convergence result establishes as a special case essentially sharp weak convergence rates for the continuous version of the hyperbolic Anderson model. Our method of proof makes use of the Kolmogorov equation and the Hölder-inequality for Schatten norms.

scholarly journals Time-Dependent Wave Equations on Graded Groups

2021 ◽  
Vol 171 (1) ◽  
Author(s):  
Michael Ruzhansky ◽  
Chiara Alba Taranto
Keyword(s):  
Wave Equation ◽  
Lie Groups ◽  
Classical Result ◽  
Wave Equations ◽  
Evolution Equations ◽  
Time Dependent ◽  
Well Posedness ◽  
Local Loss ◽  
Stratified Lie Groups ◽  
Loss Of Regularity

AbstractIn this paper we consider the wave equations for hypoelliptic homogeneous left-invariant operators on graded Lie groups with time-dependent Hölder (or more regular) non-negative propagation speeds. The examples are the time-dependent wave equation for the sub-Laplacian on the Heisenberg group or on general stratified Lie groups, or $p$ p -evolution equations for higher order operators on ${{\mathbb{R}}}^{n}$ R n or on groups, already in all these cases our results being new. We establish sharp well-posedness results in the spirit of the classical result by Colombini, De Giorgi and Spagnolo. In particular, we describe an interesting local loss of regularity phenomenon depending on the step of the group (for stratified groups) and on the order of the considered operator.

Regular solutions of wave equations containing nonlinearities of the type L Φ L

1983 ◽  
Vol 95 (1-2) ◽  
pp. 147-152 ◽  
Author(s):  
Jürgen Weyer
Keyword(s):  
Wave Equation ◽  
Elasticity Theory ◽  
Hilbert Spaces ◽  
Nonlinear Wave Equation ◽  
Selfadjoint Operator ◽  
Wave Equations ◽  
Regular Solutions ◽  
Initial Value ◽  
Abstract Result ◽  
Nonlinear Elasticity Theory

SynopsisRegular solutions of the forced nonlinear wave equation uu + L4u + LΦLu = r are studied in Hilbert spaces. L is a linear, positive, selfadjoint operator and the nonlinear nucleus Φ(u) = f(|u|2)u is generated by a C1-function f, such that LΦ(Lu) = f(|Lu|2)L2u. If the initial value data u(0) = ϕ and u1(0) = ψ belong to the domain D(Lk+4) and D(Lk+2), respectively, and if rεD(Lk), then there is a (global) solution u(t) such that u ε D(Lk+4), ut ε D(Lk+2) and uuε D(Lk) for all times t. The abstract result is applied to examples in nonlinear elasticity theory.

scholarly journals Scattering for critical wave equations with variable coefficients

2021 ◽  
pp. 1-19
Author(s):  
Shi-Zhuo Looi ◽  
Mihai Tohaneanu
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Variable Coefficients ◽  
Energy Decay ◽  
Time Dependent ◽  
Linear Wave ◽  
Decay Estimates ◽  
Local Energy ◽  
Linear Wave Equation ◽  
Semilinear Wave Equation

Abstract We prove that solutions to the quintic semilinear wave equation with variable coefficients in ${{\mathbb {R}}}^{1+3}$ scatter to a solution to the corresponding linear wave equation. The coefficients are small and decay as $|x|\to \infty$ , but are allowed to be time dependent. The proof uses local energy decay estimates to establish the decay of the $L^{6}$ norm of the solution as $t\to \infty$ .

scholarly journals DIFFERENTIAL FORMS AND WAVE EQUATIONS FOR GENERAL RELATIVITY

1998 ◽  
Vol 07 (06) ◽  
pp. 857-885 ◽  
Author(s):  
STEPHEN R. LAU
Keyword(s):  
General Relativity ◽  
Wave Equation ◽  
Degrees Of Freedom ◽  
Wave Equations ◽  
Evolution Equations ◽  
Differential Forms ◽  
Extrinsic Curvature ◽  
The Cauchy Problem ◽  
Hyperbolic Form ◽  
Tensor Index

In recent papers, Choquet–Bruhat and York and Abrahams, Anderson, Choquet–Bruhat, and York (we refer to both works jointly as AACY) have cast the 3 + 1 evolution equations of general relativity in gauge-covariant and causal "first-order symmetric hyperbolic form," thereby cleanly separating physical from gauge degrees of freedom in the Cauchy problem for general relativity. A key ingredient in their construction is a certain wave equation which governs the light-speed propagation of the extrinsic curvature tensor. Along a similar line, we construct a related wave equation which, as the key equation in a system, describes vacuum general relativity. Whereas the approach of AACY is based on tensor-index methods, the present formulation is written solely in the language of differential forms. Our approach starts with Sparling's tetrad-dependent differential forms, and our wave equation governs the propagation of Sparling's two-form, which in the "time-gauge" is built linearly from the "extrinsic curvature one-form." The tensor-index version of our wave equation describes the propagation of (what is essentially) the Arnowitt–Deser–Misner gravitational momentum.

scholarly journals ON THE NONRELATIVISTIC LIMIT OF LINEAR WAVE EQUATIONS FOR ZERO AND UNITY SPIN PARTICLES

2008 ◽  
Vol 23 (02) ◽  
pp. 129-137 ◽  
Author(s):  
P. YU. MOSHIN ◽  
J. L. TOMAZELLI
Keyword(s):  
Wave Equation ◽  
Power Series ◽  
Unitary Transformation ◽  
Wave Equations ◽  
Power Series Expansion ◽  
Particle Dynamics ◽  
Nonrelativistic Limit ◽  
Linear Wave ◽  
Linear Wave Equation ◽  
Linear Wave Equations

The nonrelativistic limit of the linear wave equation for zero and unity spin bosons of mass m in the Duffin–Kemmer–Petiau representation is investigated by means of a unitary transformation, analogous to the Foldy–Wouthuysen canonical transformation for a relativistic electron. The interacting case is also analyzed, by considering a power series expansion of the transformed Hamiltonian, thus demonstrating that all features of particle dynamics can be recovered if corrections of order 1/m2 are taken into account through a recursive iteration procedure.

scholarly journals Rotationally invariant periodic solutions of semilinear wave equations

1998 ◽  
Vol 3 (1-2) ◽  
pp. 171-180 ◽  
Author(s):  
Martin Schechter
Keyword(s):  
Wave Equation ◽  
Periodic Solutions ◽  
Essential Spectrum ◽  
Wave Operator ◽  
Wave Equations ◽  
Periodic Case ◽  
Linear Wave ◽  
Semilinear Wave Equations ◽  
Semilinear Wave Equation ◽  
Rotationally Invariant

Under suitable conditions we are able to solve the semilinear wave equation in any dimension. We are also able to compute the essential spectrum of the linear wave operator for the rotationally invariant periodic case.

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