BEM‐FEM coupling for the 1D Klein–Gordon equation

Abstract

A transmission (bidomain) problem for the one‐dimensional Klein–Gordon equation on an unbounded interval is numerically solved by a boundary element method‐finite element method (BEM‐FEM) coupling procedure. We prove stability and convergence of the proposed method by means of energy arguments. Several numerical results are presented, confirming theoretical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 2042–2082, 2014

A numerical study of energetic BEM-FEM applied to wave propagation in 2D multidomains

“…from which one can deduce a-priori stability estimates for regular solutions u 1 and u 2 bounding from above the related energies by means of the problem data [8].…”

Energetic BEM for the numerical analysis of 2D Dirichlet damped wave propagation exterior problems

“…Since wave propagation phenomena are often observed in semi-infinite media (domain) where Sommerfeld radiation condition holds, a suitable numerical method has to ensure that this condition is not violated. For example, FEMs need the application of special techniques to fulfill this condition that, on the contrary, is implicitly fulfilled by BEM; hence a suitable coupling of both these techniques, when applicable, gives undoubted advantages [17,18]. In principle, both frequency-domain and time-domain BEM can be used for hyperbolic initial-boundary value problems [19][20][21][22].…”

“…The mathematical background of time-dependent boundary integral equations is summarized by M. Costabel in [26]. For the numerical solution of the damped wave equation in 1D unbounded media, we have already considered in [17,18,27] the extension of the so-called energetic BEM, introduced for the undamped wave equation in several space dimensions [28][29][30]. The analysis carried out for 1D damped wave propagation problems allowed to fully understand the approximation technique for what concerns marching on time, avoiding space integration with BEM singular kernels and it was considered as a touchstone for the extension to higher space dimensions, which is done here for the 2D case.…”

Energetic Bem for the Numerical Solution of Damped Wave Propagation Exterior Problems

“…Consequently, the integral problem can be discretized by unconditionally stable schemes via the so-called energetic BEM. In this context, starting from the application of energetic BEM to classical wave propagation exterior problems [2,3], we consider here an extension for the damped wave equation in 2D space dimension, based on successful simulations for the 1D case [6,7]. Several benchmarks will be presented and discussed.…”

“…These latter can be obtained reformulating time-dependent problems modeled by partial differential equations (PDEs) of hyperbolic type in terms of boundary integral equations (BIEs) solved via boundary element methods (BEMs). In this context, starting from a recently developed energetic weak formulation of the space-time BIE modeling, in particular, classical wave propagation exterior problems [2,3], we consider here an extension for the damped wave equation in 2D space dimension, based on successful simulations for the 1D case [6,7]. In fact, the related energetic BEM reveals a robust time stability property, which is crucial in guaranteeing accurate numerical solutions on large time intervals.…”