About
This work is done for my PhD thesis at Cea-List and Inria Paris under the supervision of Martin Vohralík (Inria) and Alexandre Imperiale (Cea).
Context and objective
The thesis develops a posteriori error estimates for numerical simulations of the transient (acoustic and elastic) wave equation, motivated by ultrasonic non-destructive testing (NDT) applications. A priori estimates justify a scheme's convergence order, but without giving a computable bound. A posteriori estimators are fully computable from the discrete solution and can be used to certify a simulation's precision and to localise the error in space and time. The goal is to obtain estimators that are:
- Reliable (and ideally guaranteed / constant-free): an upper bound on the discretisation error.
- Locally space-time efficient: also a lower bound, so the estimator has no false positives and genuinely tracks where the error is concentrated.
- Polynomial-degree robust, where possible: with constants independent of the discretisation's polynomial order.
- Computationally cheap enough to be usable in industrial software.
We develop both residual-based and flux-based estimators, extended from the elliptic and hyperbolic settings to the space-time setting of the wave equation.
Contents
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Chapter 1: Reliable and locally space-time efficient a posteriori estimates for the transient acoustic wave equation with C¹-in-time discrete reconstructions
Considers the acoustic wave equation discretised with the spectral finite element method in space and an explicit leapfrog scheme in time. Introduces a C¹-in-time, piecewise cubic reconstruction of the leapfrog iterates into a space-time conforming solution. Derives reliable and locally space-time efficient a posteriori estimates for residual-based, averaged-flux-based, and equilibrated-flux-based estimators. Generalises these estimates into an abstract framework covering broader discretisations and anisotropic space-time meshes.
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Chapter 2: Extended two-field reconstruction and polynomial degree robust efficiency
Relaxes the regularity constraints on the discrete solution by introducing a second reconstruction field, extending the reliability and efficiency estimates of Chapter 1 to this new setting. Its main new result is polynomial-degree-robust a posteriori estimates, with constants independent of the spatial polynomial degree.
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Chapter 3: An extension by analogy to elastic waves
Extends the estimation methodology heuristically from the scalar acoustic wave equation to the elastic wave equation, relevant to NDT simulations. Validates the approach on a toy example studying uniform vs. heterogeneous material configurations, mesh refinement, and final-time effects. Shows that the error indicator roughly doubles solver run time, which is far cheaper than the brute-force alternative of a full mesh refinement used as a reference solution.
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Chapter 4: Computationally efficient a posteriori estimates in industrial software
Describes integration of the estimates into CIVA, the industrial NDT simulation software, and discusses computational efficiency considerations for practical deployment.
Publications
- Hugot, N., Imperiale, A., Vohralík, M. (2026). Reliable and locally space-time efficient a posteriori estimates for the transient acoustic wave equation. Preprint (PDF).