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 $Q^k$ 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 $\exp (C{h}^{-(1-\epsilon )})$ for any $\epsilon \in (0,1)$ 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.

arXiv comment26 pages, 2 figures. Revised and extended to multidimensional wave equations; title and author list updated