A Hybridized High-Order Method for Unique Continuation Subject to the Helmholtz Equation
A Hybridized High-Order Method for Unique Continuation Subject to the Helmholtz Equation
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DOI：
10.1137/20m1375619
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发表时间：
2020-10
期刊：
影响因子：
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通讯作者：
E. Burman;G. Delay;A. Ern
中科院分区：
文献类型：
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作者：
E. Burman;G. Delay;A. Ern
设计并分析了一种任意阶杂化不连续伽辽金方法来逼近亥姆霍兹方程下的唯一连续问题。该方法使用条件稳定性估计对连续问题进行分析，导致计算域内子域的范数误差估计。收敛阶反映了条件稳定性估计的Holder连续性和有限元空间对充分光滑解的逼近性质。在一定的凸性条件下，估计中的常数与频率无关。此外，还证明了误差的某些加权平均是收敛的，与连续问题的稳定性无关。数值算例说明了该方法在问题的不适定性程度、增加多项式阶数和数据扰动方面的性能。
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 Holder 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.