تقرير
Wavenumber-explicit convergence of the $hp$-FEM for the full-space heterogeneous Helmholtz equation with smooth coefficients
| العنوان: | Wavenumber-explicit convergence of the $hp$-FEM for the full-space heterogeneous Helmholtz equation with smooth coefficients |
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| المؤلفون: | Lafontaine, David, Spence, Euan A., Wunsch, Jared |
| سنة النشر: | 2020 |
| المجموعة: | Computer Science Mathematics |
| مصطلحات موضوعية: | Mathematics - Numerical Analysis, Mathematics - Analysis of PDEs, 35J05, 65N15, 65N30, 78A45 |
| الوصف: | A convergence theory for the $hp$-FEM applied to a variety of constant-coefficient Helmholtz problems was pioneered in the papers [Melenk-Sauter, 2010], [Melenk-Sauter, 2011], [Esterhazy-Melenk, 2012], [Melenk-Parsania-Sauter, 2013]. This theory shows that, if the solution operator is bounded polynomially in the wavenumber $k$, then the Galerkin method is quasioptimal provided that $hk/p \leq C_1$ and $p\geq C_2 \log k$, where $C_1$ is sufficiently small, $C_2$ is sufficiently large, and both are independent of $k,h,$ and $p$. The significance of this result is that if $hk/p= C_1$ and $p=C_2\log k$, then quasioptimality is achieved with the total number of degrees of freedom proportional to $k^d$; i.e., the $hp$-FEM does not suffer from the pollution effect. This paper proves the analogous quasioptimality result for the heterogeneous (i.e. variable-coefficient) Helmholtz equation, posed in $\mathbb{R}^d$, $d=2,3$, with the Sommerfeld radiation condition at infinity, and $C^\infty$ coefficients. We also prove a bound on the relative error of the Galerkin solution in the particular case of the plane-wave scattering problem. These are the first ever results on the wavenumber-explicit convergence of the $hp$-FEM for the Helmholtz equation with variable coefficients. |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2010.00585 |
| رقم الأكسشن: | edsarx.2010.00585 |
| قاعدة البيانات: | arXiv |
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