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

TL;DR

This paper provides a rigorous proof of an integration by parts formula for the hypersingular boundary integral operator related to the transient heat equation in three dimensions, clarifying the interpretation of a complex integral term.

Contribution

It offers the first complete proof of the integration by parts formula and an alternative representation of the integral term for specific function classes.

Findings

Rigorous proof of the integration by parts formula

Alternative representation of the integral term

Applicable to tensor-product discretization spaces

Abstract

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.