1. Space-time boundary element methods for the heat equation

Dohr, Stefan; Niino, Kazuki; Steinbach, Olaf

doi:10.1515/9783110548488-001

Cited by 15 publications

(19 citation statements)

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“…In this paper, we have formulated and analysed new non-standard variational formulations for finite element discretisations of parabolic and hyperbolic initial boundary value problems, in particular, for the heat and wave equations. Based on this analysis, we can analyse related boundary integral equations and boundary element methods, where we recover known results in the case of the heat equation [12], but we expect to derive new results in the case of the wave equation. Moreover, using this unified framework, it will be possible to analyse the coupling of space-time finite and boundary element methods.…”

Section: The Wave Equationmentioning

confidence: 79%

Coercive space-time finite element methods for initial boundary value problems

Steinbach¹,

Zank²

2020

etna

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We propose and analyse new space-time Galerkin-Bubnov-type finite element formulations of parabolic and hyperbolic second-order partial differential equations in finite time intervals. Using Hilbert-type transformations, this approach is based on elliptic reformulations of first-and second-order time derivatives, for which the Galerkin finite element discretisation results in positive definite and symmetric matrices. For the variational formulation of the heat and wave equations, we prove related stability conditions in appropriate norms, and we discuss the stability of related finite element discretisations. Numerical results are given which confirm the theoretical results.

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Section: The Wave Equationmentioning

confidence: 79%

Coercive space-time finite element methods for initial boundary value problems

Steinbach¹,

Zank²

2020

etna

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“…Here, we want to establish a boundary representation formula for exterior solutions to the heat equation, which uses both Dirichlet and Neumann data. Such exterior representation formula has already been mentioned in [41,50,51], but without giving an explicit growth condition on the heat equation solution that guarantees its validity. We give a growth condition for the heat equation, analogous to the Sommerfeld radiation condition for the Helhmholtz equation [52] (a comparison between these two conditions is in remark 2.5).…”

Section: Integral Representation Of Heat Equation Solutionsmentioning

confidence: 99%

“…Galerkin methods are commonly used to approximate the spatial integrals in equations (5.2), (5.1), with a number of different approaches to deal with the integration in time. For instance, time marching [38], time–space Galerkin methods [41,51], convolution quadrature [50] and collocation [59]. For simplicity, we opted for an approach based on the trapezoidal rule for the integration on normal∂Ω and the midpoint rule for the time convolution.…”

Section: A Simple Numerical Approach To Potential Theory For the Heat Equationmentioning

confidence: 99%

Active thermal cloaking and mimicking

et al. 2021

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We present an active cloaking method for the parabolic heat (and mass or light diffusion) equation that can hide both objects and sources. By active, we mean that it relies on designing monopole and dipole heat source distributions on the boundary of the region to be cloaked. The same technique can be used to make a source or an object look like a different one to an observer outside the cloaked region, from the perspective of thermal measurements. Our results assume a homogeneous isotropic bulk medium and require knowledge of the source to cloak or mimic, but are in most cases independent of the object to cloak.

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“…For more details and proofs, we refer to the seminal works [AN87, Noo88, Cos90], which considered u 0 = 0, and to [DNS19,Doh19] for the general case.…”

Section: Anisotropic Sobolev Spacesmentioning

confidence: 99%

Adaptive space-time BEM for the heat equation

Gantner,

van Venetië

2021

Preprint

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We consider the space-time boundary element method (BEM) for the heat equation with prescribed initial and Dirichlet data. We propose a residual-type a posteriori error estimator that is a lower bound and, up to weighted L 2 -norms of the residual, also an upper bound for the unknown BEM error. The possibly locally refined meshes are assumed to be prismatic, i.e., their elements are tensor-products J × K of elements in time J and space K. While the results do not depend on the local aspect ratio between time and space, assuming the scaling |J| diam(K) 2 for all elements and using Galerkin BEM, the estimator is shown to be efficient and reliable without the additional L 2 -terms. In the considered numerical experiments on two-dimensional domains in space, the estimator seems to be equivalent to the error, independently of these assumptions. In particular for adaptive anisotropic refinement, both converge with the best possible convergence rate.

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1. Space-time boundary element methods for the heat equation

Cited by 15 publications

References 0 publications

Coercive space-time finite element methods for initial boundary value problems

Coercive space-time finite element methods for initial boundary value problems

Active thermal cloaking and mimicking

Adaptive space-time BEM for the heat equation

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