2019/01/24 by Vanja Nikolić, Barbara Wohlmuth, Nikolić, Vanja +1 · 1 citation
Engineering · Mathematics · #Advanced Mathematical Physics Problems #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1901.08510
openalex publication_date 2019/01/24 · openalex created_date 2022/09/24 · openalex updated_date 2026/07/28
We study the spatial discretization of Westervelt's quasilinear strongly\ndamped wave equation by piecewise linear finite elements. Our approach employs\nthe Banach fixed-point theorem combined with a priori analysis of a linear wave\nmodel with variable coefficients. Degeneracy of the semi-discrete Westervelt\nequation is avoided by relying on the inverse estimates for finite element\nfunctions and the stability and approximation properties of the interpolation\noperator. In this way, we obtain optimal convergence rates in L2-based\nspatial norms for sufficiently small data and mesh size and an appropriate\nchoice of initial approximations. Numerical experiments in a setting of a 1D\nchannel as well as for a focused-ultrasound problem illustrate our theoretical\nfindings.\n