Sumit Mahajan (Inria RAPSODI): A Posteriori Estimates for Space Discretizations of the Wave Equation with Dynamic Meshes

This is joint work with Théophile Chaumont-Frelet (Inria Lille) and Martin Vohralík (Inria Paris). In this talk, we present fully computable a posteriori error estimates for the semidiscrete finite element approximation of the scalar wave equation on dynamically evolving spatial meshes. The error is measured in  a damped energy norm. In this norm, the estimator is reliable and globally efficient, with fully explicit constants independent of the mesh size, polynomial degree, damping parameter and frequency cutoff. Mesh transitions introduce temporal discontinuities in the discrete solution and its time derivatives, which we quantify through explicit jump indicators. These contributions are fully controlled, vanish for fixed or nested meshes, and do not compromise the robustness of the estimator. The forcing term enters through computable data-oscillation contributions, damped by the frequency split. The resulting estimator decomposes into residual, oscillation and mesh-transition terms, and reduces to the classical residual structure when the mesh does not change. Numerical experiments illustrate the theory and demonstrate the sharpness and robustness of the estimator for long-time simulations with dynamic mesh adaptation.