ABSTRACT We propose a weak Galerkin finite element method for the Westervelt's model of ultrasound waves on polygonal meshes. Specifically, we investigate the spatial discretization of Westervelt's quasi‐linear, strongly damped wave equation using high‐order weak Galerkin discretization. The primary challenges in the numerical analysis include managing the nonlinear terms in the model and preventing the equation from degenerating. We avoid the degeneracy of the semi‐discrete Westervelt's equation by employing inverse estimates and the stability and approximation properties of the projection. Our convergence analysis relies on the Banach fixed‐point theorem, along with a stability and convergence analysis of a linear diffusive viscous wave equation with variable coefficients for the first and second time derivatives. This approach yields optimal convergence rates in ‐based spatial norms for sufficiently small data and mesh size, given an appropriate choice of initial data. Numerical experiments conducted in two‐dimensional settings illustrate the theoretical convergence results.
Jana et al. (Fri,) studied this question.
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