Strong Norm Error Bounds for Quasilinear Wave Equations Under Weak CFL-Type Conditions

Foundations of Computational Mathematics https://doi.org/10.1007/s10208-024-09639-w Strong Norm Error Bounds for Quasilinear Wave Equations Under Weak CFL-Type Conditions Benjamin Dörich1 Received: 10 October 2022 / Revised: 28 October 2023 / Accepted: 31 January 2024 © The Author(s) 2024 Abstract In the present paper, we consider a class of quasilinear wave equations on a smooth, bounded domain. We discretize it in space with isoparametric finite elements and apply a semi-implicit Euler and midpoint rule as well as the exponential Euler and midpoint rule to obtain four fully discrete schemes. We derive rigorous error bounds of optimal order for the semi-discretization in space and the fully discrete methods in norms which are stronger than the classical H 1 × L 2 energy norm under weak CFL-type conditions. To confirm our theoretical findings, we also present numerical experiments. Keywords Error analysis · Full discretization · Quasilinear wave equation · Nonconforming space discretization · Isoparametric finite elements · A-priori error bounds Mathematics Subject Classification Primary: 65M12 · 65M15 · 65J15 Secondary: 65M60 · 35L05 Communicated by Arieh Iserles. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Project-ID 258734477 - SFB 1173. B 1 Benjamin Dörich Institute for Applied and Numerical Mathematics, Karlsruhe Institute of Technology, 76149 Karlsruhe, Germany 123 Foundations of Computational Mathematics 1 Introduction In the present paper, we consider the quasilinear wave equation λ(u(t, x))∂tt u(t, x) = u(t, x) + g(t, x, u(t, x), ∂t u(t, x)), (1.1) for t ∈ [0, T ], x ∈ Ω ⊂ R N , N = 1, 2, 3. We assume the domain Ω to be bounded with a sufficiently regular boundary and impose homogeneous Dirichlet boundary conditions. We discretize (1.1) in space using isoparametric finite elements and employ for the time discretization a semi-implicit Euler and midpoint rule as well as an exponential Euler and midpoint rule. We derive error bounds in norms stronger than the standard energy H 1 × L 2 -norm. The first wellposedness results for a large class of quasilinear wave type equation were given by Kato in [25, 26]. This approach was refined in [11] for the problem (1.1) to account for the state-dependent norms necessary in the analysis. A typical example in nonlinear acoustics is the model λ(u) = 1 − u m for some m ∈ N. Hence, in order to ensure λ(u) > 0, a key ingredient in the proof is to establish pointwise bounds on u (as well as ∂t u), often via Sobolev’s embedding H 2 → L ∞ . To inherit this property in the spatial discretization, we need pointwise bounds on the numerical approximations in the error analysis. However, since the finite elements space is not H 2 -conforming, we cannot follow the above approach. So far in the literature, bounds in H 1 × L 2 are shown by inverse estimates which yield a factor h −β for some β ≥ 1 with the spatial mesh width h. This induces unsatisfactory CFL-type conditions and excludes linear finite elements. In contrast with this, we adapt the idea from the wellposedness and perform the error analysis not in the energy space H 1 × L 2 , but employ a discrete version of the H 2 -norm. A discrete variant of Sobolev’s embedding and a suitably defined solution space for the numerical approximation allow us to remove lower bounds on the polynomial degree of the finite element space and significantly improve the CFL-type condition compared to the literature. For the temporal step size τ and the spatial mesh width h, we show convergence in N = 2 under the restriction τ  h α , for any α > 0, and in N = 3 we have τ  h 1/2+α for the first-order methods in time and τ  h 1/4+α for the second-order method. In addition, we fully remove the CFL-type condition for N = 1. The strategy of the semi discrete proof relies on a bootstrap argument. We set up a suitable solution space for the numerical approximation and show that the initial value lies in this. Instead of the usual choice of interpolated initial values, we have to use a Ritz map for which we provide a computable alternative of the correct order. Since we are working with a finite-dimensional subspace, this directly yields local wellposedness up to some time th∗ > 0. On this possibly short time interval, we prove convergence in the stronger norm and use this to extend th∗ beyond T and to close the argument. For the fully discrete error bounds, this approach is generalized using an induction argument. We give a brief overview of the literature on the numerical treatment of quasilinear wave equations. In the pioneering works [10, 24, 27, 37], existence of solutions to quasilinear and nonlinear evolution equations is established, and one can find approx- 123 Foundations of Computational Mathematics imation rates of the implicit and semi-implicit Euler method. Within an (extended) Kato framework, optimal order for these methods was achieved in [22] and rigorous error bounds for the time discretization by higher-order Runge–Kutta methods are derived in [23, 28]. Concerning the spatial discretization, the results in [21] yield optimal order of convergence for the equation (1.1), however, only for polynomials of degree greater than two. For the strongly damped Westervelt equation, continuous and discontinuous Galerkin methods were analyzed in [1, 34]. Very recently, mixed finite elements for the Kuznetsov and Westervelt equations were studied in [33]. In [31], error bounds for two variant of the midpoint rule are derived of optimal order, but only for polynomials of degree greater than two and under a stronger CFLtype condition compared to our results. In the case of one-dimensional wave equation subject to periodic boundary conditions, full discretization error bounds are established in [19]. A sophisticated energy technique combined with the properties of the spectral discretization yields convergence without a CFL-type condition. For a slightly different quasilinear wave equation, optimal error bounds in L 2 for continuous finite elements were considered in the literature. One-step methods of different order are analyzed in [3, 4, 17], and two-step methods are considered in [5]. For a class of linearly implicit single-step schemes as well as a linearly and a fully implicit two-step scheme, optimal error bounds are derived in [32]. However, all of these results require a CFL-type condition at least as strong as τ  h and do not allow for linear finite elements. We expect that our technique can be generalized to these problems, but this will be part of future research. The paper is organized as follows: We describe in Sect. 2 the analytical framework, the space discretization by isoparametric Lagrange finite element, and state our main results. The proof of the spatial convergence rates is given in Sect. 3, where we first reduce the main result to error bounds in a stronger energy norm which is established afterward. In Sect. 4, we extend (...truncated)