A Priori Error Estimates for the Finite Element Approximation of Westervelt's Quasi-linear Acoustic Wave Equation

Abstract

We study the spatial discretization of Westervelt's quasilinear strongly damped wave equation by piecewise linear finite elements. Our approach employs the Banach fixed-point theorem combined with a priori analysis of a linear wave model with variable coefficients. Degeneracy of the semi-discrete Westervelt equation is avoided by relying on the inverse estimates for finite element functions and the stability and approximation properties of the interpolation operator. In this way, we obtain optimal convergence rates in L 2 -based spatial norms for sufficiently small data and mesh size and an appropriate choice of initial approximations. Numerical experiments in a setting of a 1D channel as well as for a focused-ultrasound problem illustrate our theoretical findings.

A stabilizer free weak Galerkin finite element method for second‐order Sobolev equation

“…The following result deals with the existence and uniqueness of the SFWG solution $$$ {u}_h. $$$ The basic technique is borrowed from [33]. Theorem For each $$$ h\in \left(0,{h}_0\right], $$$ there exist a function $$$ {u}_h\in {C}^1\left(0,T;{\mathcal{V}}_h^0\right) $$$ satisfying (3.1).…”

“…The following result deals with the existence and uniqueness of the SFWG solution u h . The basic technique is borrowed from [33].…”

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

“…To the best of our knowledge, this is the first such result in the context of nonlinear and nonlocal acoustic equations. In terms of the closely related works, we point out the non-uniform finite element analysis of the local Westervelt equation (with K = δ 0 and ε > 0) in [25]. Finite element analysis of the inviscid problem (K = δ 0 and ε = 0) follows as a special case of the results in [6,11].…”

Exponential Integrators for Quasilinear Wave-Type Equations

“…These bound were shown for a trigonometric integrator by a sophisticated stability analysis. In [2,27] continuous and discontinuous Galerkin (dG) methods were used for the space discretization of the Westervelt equation in two and three dimensions and absorbing boundary conditions for this equation were treated in [25]. Concerning full discretization, we further mention the work [5], where error bounds for linear finite elements in space combined with a dG method of order 0 in time for parabolic problems were derived under low regularity assumptions.…”