Error analysis of the DtN-FEM for the scattering problem in acoustics via Fourier analysis

Abstract

a b s t r a c tIn this paper, we are concerned with the error analysis for the finite element solution of the two-dimensional exterior Neumann boundary value problem in acoustics. In particular, we establish explicit priori error estimates in H 1 and L 2 -norms including both the effect of the truncation of the DtN mapping and that of the numerical discretization. To apply the finite element method (FEM) to the exterior problem, the original boundary value problem is reduced to an equivalent nonlocal boundary value problem via a Dirichlet-toNeumann (DtN) mapping represented in terms of the Fourier expansion series. We discuss essential features of the corresponding variational equation and its modification due to the truncation of the DtN mapping in appropriate function spaces. Numerical tests are presented to validate our theoretical results.

An efficient iterative method for solving multiple scattering in locally inhomogeneous media

“…n (κRij )v n e inθij with a suitable cut-off number N. Harari and Hughes [31] showed that the choice of N ≥ κR ij could guarantee the solvability of the approximate problem with a certified accuracy. We also refer to [32] for the error analysis and numerical studies on the selection of an optimal cut-off number N. In practice, the choice N ≥ κR ij is always safe although it is conservative at times. Grote and Keller [33] suggested a different modification of the DtN boundary condition to remove the constraint on κR ij for any fixed N.…”

Convergence of an adaptive finite element DtN method for the elastic wave scattering by periodic structures

“…Practically, the infinite series needs to be truncated into the sum of finite number of terms by choosing an appropriate truncation parameter N . It is known that the convergence of the truncated DtN map could be arbitrarily slow to the original DtN map in the operator norm [35]. To overcome this issue, the duality argument has to be developed to obtain the a posteriori error estimate between the solution of the scattering problem and the finite element solution.…”

A BIE-Based DtN-FEM for Fluid-Solid Interaction Problems

“…In particular, we can derive a Dirichlet-to-Neumann (DtN) mapping on the artificial boundary to obtain an exact transparent boundary condition, and accordingly this strategy is called DtN method [13]. The DtN mapping can be computed by boundary integral operators [18,23,32] or by Fourier-series expansions [12,38]. The boundary integral equation based (BIE-based) DtN mapping can be defined on any smooth closed curve and this feature may reduce the size of the computational domain, while the Fourier expansion series based DtN mapping is usually defined on a circle or on a perturbation of a circle [34].…”