Asymptotic-preserving finite element analysis of Westervelt-type wave equations

Motivated by numerical modeling of ultrasound waves, we investigate conforming finite element approximations of quasilinear and possibly nonlocal equations of Westervelt type. These wave equations involve either a strong dissipation or damping of fractional-derivative type and we unify them into one class by introducing a memory kernel that satisfies non-restrictive regularity and positivity assumptions. As the involved damping parameter is relatively small and can become negligible in certain (inviscid) media, it is important to develop methods that remain stable as the said parameter vanishes. To this end, the contributions of this work are two-fold. First, we determine sufficient conditions under which conforming finite element discretizations of (non)local Westervelt equations can be made robust with respect to the dissipation parameter. Second, we establish the rate of convergence of the semi-discrete solutions in the singular vanishing dissipation limit. The analysis hinges upon devising appropriate energy functionals for the semi-discrete solutions that remain uniformly bounded with respect to the damping parameter. Numerical experiments are provided to illustrate the theory.