Unique continuation for the wave equation based on a discontinuous Galerkin time discretization

TL;DR

This work addresses stable unique continuation for the wave equation with incomplete initial data by formulating and analyzing two discretizations: a semi-discrete conforming FEM and a fully-discrete discontinuous Galerkin-in-time FEM, both combined with continuous spatial FE. It provides rigorous error estimates under the geometric control condition and develops practical preconditioning strategies to solve the resulting globally coupled space-time systems using time-stepping, thereby enabling computations in three spatial dimensions without full space-time meshing. The key contributions are the identification of global couplings inherent to achieving optimal convergence, the construction of stable augmented Lagrangian formulations with carefully designed stabilization, and two tractable preconditioning approaches (monolithic time-marching and decoupled forward-backward) that leverage slab-wise solves. Numerical experiments validate the proposed methods, demonstrate convergence consistent with theory, and illustrate the critical role of GCC for accurate recovery outside the data domain, highlighting the method’s potential for data assimilation in wave propagation problems.

Abstract

We consider a stable unique continuation problem for the wave equation where the initial data is lacking and the solution is reconstructed using measurements in some subset of the bulk domain. Typically fairly sophisticated space-time methods have been used in previous work to obtain stable and accurate solutions to this reconstruction problem. Here we propose to solve the problem using a standard discontinuous Galerkin method for the temporal discretization and continuous finite elements for the space discretization. Error estimates are established under a geometric control condition. We also investigate two preconditioning strategies which can be used to solve the arising globally coupled space-time system by means of simple time-stepping procedures. Our numerical experiments test the performance of these strategies and highlight the importance of the geometric control condition for reconstructing the solution beyond the data domain.

Figures (4)

• Figure 1: Results for one-dimensional experiment of \ref{['ssection:precond-1d']} using $h = \Delta t$. The upper panel refers to $q=k=1$ while the lower panel uses $q=k=2$. The tables contain iteration numbers for GMRes preconditioned by the methods proposed in \ref{['section:precond']} using tolerance $10^{-7}$. The column 'ndof' gives the total number of degrees of freedom for the 'M-f'-method. The two plots on the right show convergence of $\left\lVert\partial_t \tilde{e}_h \right\rVert_{L^{2}(0,T;L^2(\Omega))}$ with $\tilde{e}_h := u - L_{\Delta t} \underline{u}_1$. In the history of the residual only every fourth iteration is shown as a data point.
• Figure 2: Convergence plots for the unit cube with data given in $\omega = \Omega \setminus [1/4,3/4]^3$. Results are given for the 'M-l' and 'DFB' methods. The central figure displays the absolute error on the indicated refinement level.
• Figure 3: On top: Convergence plots for the unit cube with data given in merely in $\omega = [0,1/4] \times [0,1]^2$. Results are given for the 'M-l' and 'DFB' methods. Bottom: GMRes iteration numbers for $q=k=2$.
• Figure 4: Results for $\Omega = [0,1]$ with data given in $[0,1/4]$ computed with the 'M-f' method. On the left: $\left\lVert u - \mathcal{L}_{\Delta t} \underline{u}_1 \right\rVert_{L^{\infty}(0,T;L^2(\Omega))}$ (solid lines) and $\left\lVert u - \mathcal{L}_{\Delta t} \underline{u}_1 \right\rVert_{L^{\infty}(0,T;L^2(B_t))}$ (dashed lines) for the restricted set $B_t$ defined in \ref{['def:Bt']}. Similarly on the right for the time derivative in the $L^2-L^2$-norm.

Theorems & Definitions (40)

• Theorem 1.1: Lipschitz stability
• Proof 1
• Lemma 2.1
• Proof 2
• Corollary 2.2
• Proof 3
• Lemma 2.3: Continuity
• Proof 4
• Lemma 2.4: Interpolation
• Lemma 2.5