2024/12/12 by Sergio Gómez, Gómez, Sergio, Vanja Nikolić +1 · 1 citation
Engineering · #35L72 #65M12 #65M60 #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in engineering
paper · pdf · doi:10.48550/arxiv.2412.09095
openalex publication_date 2024/12/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We propose and analyze a space-time finite element method for Westervelt's quasilinear model of ultrasound waves in second-order formulation. The method combines conforming finite element spatial discretizations with a discontinuous-continuous Galerkin time stepping. Its analysis is challenged by the fact that standard Galerkin testing approaches for wave problems do not allow for bounding the discrete energy at all times. By means of redesigned energy arguments for a linearized problem combined with Banach's fixed-point argument, we show the well-posedness of the scheme, a priori error estimates, and robustness with respect to the strong damping parameter δ. Moreover, the scheme preserves the asymptotic preserving property of the continuous problem; more precisely, we prove that the discrete solutions corresponding to δ>0 converge, in the singular vanishing dissipation limit, to the solution of the discrete inviscid problem. We use several numerical experiments in (2 + 1) dimensions to validate our theoretical results.