Preconditioned Space-Time Boundary Element Methods for the One-Dimensional Heat Equation

Dohr, Stefan; Steinbach, Olaf

doi:10.1007/978-3-319-93873-8_22

Cited by 8 publications

(20 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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: (B) Reproducing Fields Exterior To a Bounded Regionmentioning

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 ∂Ω and the midpoint rule for the time convolution.…”

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

confidence: 99%

See 1 more Smart Citation

Active thermal cloaking and mimicking

et al. 2021

View full text Add to dashboard Cite

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.

show abstract

Section: (B) Reproducing Fields Exterior To a Bounded Regionmentioning

confidence: 99%

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

confidence: 99%

Active thermal cloaking and mimicking

et al. 2021

View full text Add to dashboard Cite

show abstract

“…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

View full text Add to dashboard Cite

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.

show abstract

“…two space dimensions plus the additional time dimension, is provided together with an outline of its proof. A formula for the 3+1D case can be found for example in [5] and [14], but to the best of our knowledge no proof is provided in the literature. In addition, the formula in the mentioned works contains a boundary integral including the time derivative of the fundamental solution of the heat equation, which per se is locally not integrable on the considered integration domain.…”

Section: Introductionmentioning

confidence: 99%

“…The aforementioned 'time derivative term' of this formula, will be further investigated in Section 5. Here we will give the details why the formulation in [5,14] is not adequate in general and provide our alternative in Theorem 5.2. Section 6 concludes the paper with a short summary and outlook.…”

Section: Introductionmentioning

confidence: 99%

An integration by parts formula for the bilinear form of the hypersingular boundary integral operator for the transient heat equation in three spatial dimensions

Watschinger¹,

Of²

2021

Preprint

View full text Add to dashboard Cite

While an integration by parts formula for the bilinear form of the hypersingular boundary integral operator for the transient heat equation in three spatial dimensions is available in the literature, a proof of this formula seems to be missing. Moreover, the available formula contains an integral term including the time derivative of the fundamental solution of the heat equation, whose interpretation is difficult at second glance. To fill these gaps we provide a rigorous proof of a general version of the integration by parts formula and an alternative representation of the mentioned integral term, which is valid for a certain class of functions including the typical tensor-product discretization spaces.

show abstract

Preconditioned Space-Time Boundary Element Methods for the One-Dimensional Heat Equation

Cited by 8 publications

References 7 publications

Active thermal cloaking and mimicking

Active thermal cloaking and mimicking

Adaptive space-time BEM for the heat equation

An integration by parts formula for the bilinear form of the hypersingular boundary integral operator for the transient heat equation in three spatial dimensions

Contact Info

Product

Resources

About