دانلود رایگان مقاله تحلیل خطای پیشینی و پسینی برای معادلات وابسته به زمان ماکسول

1. Introduction

Wave propagation phenomena happen in a broad range of applications. Examples include sound waves, light waves and water waves, which arise in acoustics, electromagnetics, and fluid dynamics, respectively. The wave propagation problem is often described by the second-order hyperbolic equation, also called the wave equation. Upon considering time-harmonic (steady-state) waves, the wave equation reduces to the Helmholtz equation. Over the last four decades there has been considerable interest in developing various finite element methods (FEMs) for solving the wave equation (e.g., [1–6], and references therein). To solve the wave equation more efficiently, adaptive FEMs are often used. Adaptive FEMs are often based on a posteriori error estimates, i.e., some computable quantities that estimate the FEM solution error in a suitable norm. Over the last three decades many a posteriori error estimates have been developed for time-independent problems such as elliptic equations and Helmholtz equation (e.g., [7–13], and references therein). As for time-dependent prob-lems, considerable results on a posteriori error estimates for parabolic problems have also been obtained (e.g., [14–17], and references therein). However, according to Bernardi and Suli back in 2005 [ ¨ 18, p. 199]: “the a posteriori error analysis of finite element approximations to second-order hyperbolic problems is in a less complete state”. Indeed, compared to results on elliptic, parabolic, and first order hyperbolic problems, “hyperbolic problems of second order have been much less studied” as Picasso mentioned in 2010 [19, p. 2213]. Though some a posteriori error estimates have been obtained over the years (e.g., [20–22,18,19]), “the design and implementation of adaptive algorithms for the wave equation based on rigorous a posteriori error estimators is a largely unexplored subject” as Georgoulis, Lakkis and Makridakis remarked in their 2013 paper [23, p. 1262].

5. Conclusions

In this paper, we initiate the study of a posteriori error estimates for time-dependent Maxwell’s equations. Estimators are obtained for both semi and fully discrete schemes. Though our proofs are given for ϵ = µ = 1, we believe that similar results can be directly proved for variable parameters ϵ and µ if they are bounded below and above in Ω. Further derivation of other error estimators for more general cases (including other type boundary conditions) will be explored. The numerical implementation of the a posteriori error estimators for time-dependent Maxwell’s equations will be considered in the future, since our past works on wave propagation simulation in metamaterials [30] showed that the adaptive finite element method seems necessary for 3D time-domain cloaking simulation [39].