In time-dependent boundary integral equations, a boundary element method in space can be coupled with a different type of discretization in time. For the latter a procedure based on linear multistep methods is proposed, which is applicable whenever the Laplace transform of the fundamental solution is known. The stability properties of the method are obtained from those of the underlying multistep method. In the absence of a space discretization, the numerical solution given by the proposed method is identical to that of a semi-discretization in time of the partial differential equation by the underlying multistep method. The theory is presented for the single layer potential equation of the heat equation. Convergence estimates, which are pointwise in time and expressed in terms of the boundary data, are obtained for full discretizations using Galerkin or collocation boundary element methods in space. Numerical examples are included.

/pdf/time-discretization-of-parabolic-boundary-integral-equations-3pjy4syu9k.pdf

Time discretization of parabolic boundary integral equations

Kirchhoff?s formula is the equivalent to Green?s Third Theorem for transient
waves, representing the solution of the three dimensional wave equation in terms
of its boundary data. In this classical result two retarded layer potentials appear.
We show in this paper a precise description of these potentials as time convolution
with adequate tempered distributions with values on operator spaces. With these
potentials in hands we give a self?contained proof of the formula with minimal
smoothness requirements.

A distributional version of Kirchhoff¿s formula

Topics in Functional Analysis and Applications

This thesis studies boundary integral methods for solving time-dependent partial differential equations from continuum mechanics. The two models we analyze will be transient Stokes flow and scalar acoustic scattering by penetrable obstacles. We will see the two main flavors of analysis of Time Domain Boundary Integral Equations in these two problems: for analysis of the Stokes system we take a Laplace domain approach that dates to [7]. The analysis of the acoustic scattering and transmission problem will be carried out with the newer semigroup theory based analysis from [73] and [36]. We begin with a detailed exposition of a central tool, Convolution Quadrature, that we will use throughout the rest of the thesis for temporal discretization. We provide motivation for the method from various points of view and derive both multistep and Runge-Kutta Convolution Quadrature with an eye towards implementation of the method. From this foundation, we move on to the analysis of transient Stokes flow by way of the Laplace transform. This leads us to a detailed study the Brinkman equations. Analysis of the Brinkman Single Layer potential and operator is then used to derive stability and convergence results back in the time domain for the Stokes problem. We provide numerical experiments and simulations using various spatial discretization schemes. Finally, we study the scattering of acoustic waves by inhomogeneous penetrable obstacles. Chapter 4 presents a detailed stability and convergence analysis of a three-field boundary and finite element coupling scheme. By showing the underlying problem generates a C0-group of isometries, we are able to prove stability and convergence of the scheme directly in the time domain. In a similar vein, Chapter 5 explores computationally two alternative coupling schemes.

Some applications of integral equations to the solution of transient partial differential equations

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Some applications of integral equations to the solution of transient partial differential equations