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A Hybridized High-Order Method for Unique Continuation Subject to the Helmholtz Equation
SIAM Journal on Numerical Analysis ( IF 2.9 ) Pub Date : 2021-09-14 , DOI: 10.1137/20m1375619
Erik Burman , Guillaume Delay , Alexandre Ern

SIAM Journal on Numerical Analysis, Volume 59, Issue 5, Page 2368-2392, January 2021.
We design and analyze an arbitrary-order hybridized discontinuous Galerkin method to approximate the unique continuation problem subject to the Helmholtz equation. The method is analyzed using conditional stability estimates for the continuous problem, leading to error estimates in norms over interior subdomains of the computational domain. The convergence order reflects the Hölder continuity of the conditional stability estimates and the approximation properties of the finite element space for sufficiently smooth solutions. Under a certain convexity condition, the constant in the estimates is independent of the frequency. Moreover, certain weighted averages of the error are shown to converge independently of the stability properties of the continuous problem. Numerical examples illustrate the performances of the method with respect to the degree of ill-posedness of the problem, increasing polynomial order, and perturbations in the data.


中文翻译:

服从亥姆霍兹方程的唯一连续的混合高阶方法

SIAM 数值分析杂志,第 59 卷,第 5 期,第 2368-2392 页,2021 年 1 月。
我们设计并分析了一种任意阶混合不连续伽辽金方法来逼近 Helmholtz 方程的唯一连续问题。该方法使用连续问题的条件稳定性估计进行分析,导致计算域内部子域上范数的误差估计。收敛阶数反映了条件稳定性估计的 Hölder 连续性和有限元空间的近似特性,以获得足够平滑的解。在一定的凸性条件下,估计中的常数与频率无关。此外,误差的某些加权平均值显示为独立于连续问题的稳定性属性收敛。
更新日期:2021-09-14
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