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A posteriori error analysis for the conical diffraction problem. (English) Zbl 1393.78007

Summary: Maxwell’s equations for conical diffraction can be reduced to a system of two Helmholtz equations in \(\mathbb R^2\) coupled via quasi-periodic transmission conditions on a set of piecewise smooth interfaces. A finite element formulation of the conical diffraction problem is presented in a bounded domain by introducing the nonlocal boundary operators. An a posteriori error estimate is established when the truncation of the nonlocal boundary operators takes place. As our work in [Z.-F. Wang et al., SIAM J. Numer. Anal. 53, No. 3, 1585–1607 (2015; Zbl 1328.65249)], a duality argument is applied to overcome the difficulty caused by the fact that the truncated pseudo-differential mapping does not converge to the original pseudo-differential mapping in its operator norm. The a posteriori error estimate consists of two parts: finite element discretization error and the truncation error of the nonlocal boundary operators. In particular, the truncation error is exponentially decaying with respect to the truncation parameter.

MSC:

78A45 Diffraction, scattering
78A25 Electromagnetic theory (general)
35Q60 PDEs in connection with optics and electromagnetic theory
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs

Citations:

Zbl 1328.65249
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References:

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