For the $h$-finite-element method ($h$-FEM) applied to the Helmholtz equation, the question of how quickly the meshwidth $h$ must decrease with the frequency $k$ to maintain accuracy as $k$ increases has been studied since the mid 80's. Nevertheless, there still do not exist in the literature any $k$-explicit bounds on the relative error of the FEM solution (the measure of the FEM error most often used in practical applications), apart from in one dimension. The main result of this paper is the sharp result that, for the lowest fixed-order conforming FEM (with polynomial degree, $p$, equal to one), the condition "$h^2 k^3$ sufficiently small" is sufficient for the relative error of the FEM solution in 2 or 3 dimensions to be controllably small (independent of $k$) for scattering of a plane wave by a nontrapping obstacle and/or a nontrapping inhomogeneous medium. We also prove relative-error bounds on the FEM solution for arbitrary fixed-order methods applied to scattering by a nontrapping obstacle, but these bounds are not sharp for $p\geq 2$. A key ingredient in our proofs is a result describing the oscillatory behaviour of the solution of the plane-wave scattering problem, which we prove using semiclassical defect measures.
翻译:对于适用于Helmholtz方程式的美元-freite-element 方法(美元-FEM),对于适用于Helmholtz方程式的美元-finite-element 方法,自80年代中期以来研究过,为了保持准确性,对美元-form$的增加进行了研究,因此,Meshwith-front-front-front-front-front-front-front-front-front-front-front-front-front-front-front-front-front-front-front-front-front-front-front-front-front-front-front-front-front-formal-formalld-formalld-formation-formall-formation-prolation rolation maismet-motion romotion romaismismismism)方法,我们在FEM2 k2 k2k-de-de-stmet-tolation-tomlation-tomet-tomet-tomaismaismaismismismad-tomismaist-tomis rois roismaismlation roismaism),使用了一种不制式式式式的方法方法,这种不透明式的方法,这种不使用这种不制式式的硬式的硬-de的方法,这种不制制式的方法,这是一种不制制制式式式的方法,但正正正正正正正正正正正制式方法。。。