Time‐Dependent Problems with the Boundary Integral Equation Method

Abstract

Time-dependent problems that are modeled by initial-boundary value problems for parabolic or hyperbolic partial differential equations can be treated with the boundary integral equation method. The ideal situation is when the right-hand side in the partial differential equation and the initial conditions vanish, the data are given only on the boundary of the domain, the equation has constant coefficients, and the domain does not depend on time. In this situation, the transformation of the problem to a boundary integral equation follows the same well-known lines as for the case of stationary or time-harmonic problems modeled by elliptic boundary value problems. The same main advantages of the reduction to the boundary prevail: Reduction of the dimension by one, and reduction of an unbounded exterior domain to a bounded boundary.There are, however, specific difficulties due to the additional time dimension: Apart from the practical problems of increased complexity related to the higher dimension, there can appear new stability problems. In the stationary case, one often has unconditional stability for reasonable approximation methods, and this stability is closely related to variational formulations based on the ellipticity of the underlying boundary value problem. In the timedependent case, instabilities have been observed in practice, but due to the absence of ellipticity, the stability analysis is more difficult and fewer theoretical results are available.In this article, the mathematical principles governing the construction of boundary integral equation methods for time-dependent problems are presented. We describe some of the main algorithms that are used in practice and have been analyzed in the mathematical literature.

Solvability of a Dirichlet problem for a time fractional diffusion-wave equation in Lipschitz domains

“…the constant in the error estimate blows up as Re s → 0. The same phenomenon occurs also in the case of the wave equation as indicated in [4].…”

Time Domain Boundary Element Methods for the Neumann Problem: Error Estimates and Acoustic Problems

“…Space-time anisotropic Sobolev spaces on the boundary Γ provide a convenient setting to study the mapping properties of the time-dependent layer operators [3,13]. We more generally consider the case of a screen, where the orientable, (d − 1)-dimensional Lipschitz submanifold Γ ⊂ R d may have a boundary.…”

Time‐dependent fluid‐structure interaction

“…One may show as in classical potential theory [17] that the Cauchy data φ and λ at smooth points of Γ are related by the system of boundary integral equations form the form (see, e.g., [2,3,5,11])…”

“…They are not only nonlocal in space but also nonlocal in time. As pointed out in [5], it is not clear how to choose appropriate function spaces because of the retarded argument. On the other hand, it is known that for the long time behavior of the solution, one may replace the nonlocal differential boundary integral equation by an appropriate approximated transparent boundary condition (see, e.g.…”