Time-domain boundary integral equation modeling of heat transmission problems

Abstract

This paper investigates the numerical modeling of a time-dependent heat transmission problem by the convolution quadrature boundary element method. It introduces the latest theoretical development into the error analysis of the numerical scheme. Semigroup theory is applied to obtain stability in spatial semidiscrete scheme. Functional calculus is employed to yield convergence in the fully discrete scheme. In comparison to the traditional Laplace domain approach, we show our approach gives better estimates.

Runge–Kutta approximation for $$C_0$$-semigroups in the graph norm with applications to time domain boundary integral equations

“…We only briefly give the mathematical setting. More details and a more involved physical example can be found in [27]. The setting is as follows: find u : In order to derive the boundary integral formulation, we take the Laplace transform of (8.21a), giving for j :¼ ffiffi s p :…”

“…We note that the applications in this section are chosen for their simplicity. More complicated applications, also involving full discretizations by convolution quadrature and boundary elements of systems of time domain boundary integral equations can be found in [ 29 ] and [ 27 ].…”

“…We only briefly give the mathematical setting. More details and a more involved physical example can be found in [ 27 ]. The setting is as follows: find such that It is well-known that with homogeneous Dirichlet boundary conditions generates an analytic semigroup (see e.g.…”

Active thermal cloaking and mimicking

“…2, §3, theorem 7) (see also remark 2.9) or via Sobolev regularity estimates in space and time [41,49]. For the transmission problem, growth restrictions in the Laplace domain are used to prove uniqueness in [50]. Here, we want to establish a boundary representation formula for exterior solutions to the heat equation, which uses both Dirichlet and Neumann data.…”

“…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).…”