Space-time finite element methods for nonlinear wave equations via elliptic regularisation

TL;DR

The paper develops a conforming space-time Galerkin method for the defocusing semilinear wave equation by embedding it in De Giorgi's elliptic regularisation framework, yielding an elliptic-in-time variational problem with a parameter $\varepsilon$. It proves well-posedness and unconditional stability of the discrete scheme, derives a priori error estimates in weighted $\varepsilon$-norms, and shows quasi-optimality under general discretisations; convergence rates are obtained for small nonlinearities and balanced with the regularisation. The authors address practical issues such as conditioning and underflow in the presence of the exponential weight, propose preconditioning strategies, and validate the theory with numerical experiments in one dimension, including linear and nonlinear cases. The results demonstrate that the approach provides a viable and robust framework for locally refined, high-order space-time discretisations of nonlinear wave phenomena, with potential extensions to more complex dispersive problems under the WIDE paradigm. Overall, the work offers a principled route to stable, high-fidelity simulations of nonlinear waves using space-time finite elements grounded in elliptic regularisation.

Abstract

We present and analyse a new conforming space-time Galerkin discretisation of a semi-linear wave equation, based on a variational formulation derived from De Giorgi's elliptic regularisation viewpoint of the wave equation in second-order formulation. The method is shown to be well-posed through a minimisation approach, and also unconditionally stable for all choices of conforming discretisation spaces. Further, a priori error bounds are proven for sufficiently smooth solutions. Special attention is given to the conditioning of the method and its stable implementation. Numerical experiments are provided to validate the theoretical findings.

Figures (6)

• Figure 1: The space-time cylinder $D$ (\ref{['fig: space-time cyclinder']}) and its subdivision into space-time slabs (\ref{['fig: space-time cyclinder with slabs']}).
• Figure 2: Illustration of the spatial triangulation $\mathcal{T}_{h}$ on the space-time slab $\Omega\times I_i$ in \ref{['fig: individual slab']} and the individual prismatic elements in \ref{['fig: individual element']}.
• Figure 3: The condition number of the matrix in \ref{['eq: exp linear system']}, depicting $\kappa(\varepsilon^2\tilde{K}_{\tau}+\lambda \tilde{M}_{\tau})$ against $\varepsilon$ for fixed $\tau=2^{-6}$ with $T=2$, for various values of $\lambda =1,1000,1000000$.
• Figure 4: Convergence history in the parameters $\tau,h$ to the linear, regularised problem with solution $u^{\varepsilon}$, for $\varepsilon = 2^{-2}$ fixed. In \ref{['fig: linear tau']}, $h=2^{-8}$ is fixed with $\tau$ decreasing. In \ref{['fig: linear space']}, $\tau=2^{-5}$,is fixed with decreasing $h$.
• Figure 5: Two plots indicating the convergence rates in the parameters $\tau,h$ to the nonlinear regularised $u^{\varepsilon}$ problem with $\varepsilon = 2^{-2}$ fixed. In \ref{['fig: nonlinear tau']}, $h=2^{-8}$ is fixed with $\tau$ decreasing. In \ref{['fig: nonlinear space']}, $\tau=2^{-5}$,is fixed with decreasing $h$.

Theorems & Definitions (27)

• Conjecture 2.1: De Giorgi de1996congetture
• Theorem 2.2
• proof
• Remark 2.3
• Remark 2.4
• Remark 3.1
• Remark 3.2
• Theorem 4.1: Quasi-optimality
• proof
• Remark 4.2