Asymptotic-preserving analysis and finite element discretization of nonlinear acoustic phenomena

Research in nonlinear acoustics is fueled by a growing number of high-intensity ultrasound applications in medicine and industry.  At high intensities or frequencies, sound evolution is quasilinear, and in tissue like-media, additionally nonlocal effects of time-fractional type come into play.  This application field gives rise to many interesting mathematical questions involving such nonlinear and nonlocal wave equations, including discretization under realistic (smallness) assumptions on the data and dealing with singular behavior in the vanishing limit of medium parameters such as the sound diffusivity and thermal relaxation time. In this talk, we will give an overview of these questions and then present some of our recent work on the robust mathematical and numerical analysis of singularly perturbed nonlinear acoustic wave equations. We will focus on the analysis of conforming finite element discretizations in space in the latter context, and acoustic equations of Westervelt-type with (non)local dissipation. Here we will draw parallels from their uniform treatment in the continuous setting, as both rest upon devising appropriate energy functionals that remain stable in the zero dissipation or relaxation limit.