Uniform Observability of High-Order Fully Discrete Multidimensional Wave Equations: Exponential Loss and Recovery
Abstract
Stable discrete approximation of wave controls requires observability estimates that remain uniform under mesh refinement. Such uniformity is known to fail for many low-order discretizations, but the quantitative understanding of observability loss for arbitrary high-order approximations is largely open, owing to their multiple spectral branches and the absence of explicit dispersion relations. To address this gap, we establish a quantitative theory of observability loss and recovery for fully discrete Qk local discontinuous Galerkin approximations of the multidimensional wave equation, valid for any fixed polynomial degree k. Our analysis of the (k+1)d-branch discrete spectrum identifies a unique physical branch that recovers the continuous propagation speed at low frequencies. Under a strict stability margin on this branch, its group velocity vanishes at high-frequency symmetry points. To quantify the resulting loss of uniform observability, we prove that the observability constant satisfies an exponential lower bound of the form exp(Ch−(1−ε)) for any ε∈(0,1) by constructing Gevrey-localized wave packets. We then provide a recovery strategy that restores uniform observability for any fixed polynomial degree k, with a sufficient observation time controlled by the minimum retained physical group velocity. Finally, our numerical results suggest that higher-order approximations retain a broader genuine physical-frequency band at a prescribed positive group-speed threshold, allowing less restrictive filtering in wave-control computation and better preserving wave dynamics.
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BibTeX
@misc{li2026uniform,
title = {{Uniform Observability of High-Order Fully Discrete Multidimensional Wave Equations: Exponential Loss and Recovery}},
author = {Yunzhang Li and Xiaoyang Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.17464},
eprint = {2605.17464},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.17464},
}
RIS
TY - UNPB
TI - Uniform Observability of High-Order Fully Discrete Multidimensional Wave Equations: Exponential Loss and Recovery
AU - Yunzhang Li
AU - Xiaoyang Wang
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2605.17464
UR - https://arxiv.org/abs/2605.17464
AB - Stable discrete approximation of wave controls requires observability estimates that remain uniform under mesh refinement. Such uniformity is known to fail for many low-order discretizations, but the quantitative understanding of observability loss for arbitrary high-order approximations is largely open, owing to their multiple spectral branches and the absence of explicit dispersion relations. To address this gap, we establish a quantitative theory of observability loss and recovery for fully discrete Q^k local discontinuous Galerkin approximations of the multidimensional wave equation, valid for any fixed polynomial degree k. Our analysis of the (k+1)^d-branch discrete spectrum identifies a unique physical branch that recovers the continuous propagation speed at low frequencies. Under a strict stability margin on this branch, its group velocity vanishes at high-frequency symmetry points. To quantify the resulting loss of uniform observability, we prove that the observability constant satisfies an exponential lower bound of the form exp(Ch^−(1−ε)) for any ε∈(0,1) by constructing Gevrey-localized wave packets. We then provide a recovery strategy that restores uniform observability for any fixed polynomial degree k, with a sufficient observation time controlled by the minimum retained physical group velocity. Finally, our numerical results suggest that higher-order approximations retain a broader genuine physical-frequency band at a prescribed positive group-speed threshold, allowing less restrictive filtering in wave-control computation and better preserving wave dynamics.
ER -