A finite element exponential integrator for rough solutions of semilinear wave equations. Part I: Dirichlet boundary conditions

We study a fully discrete scheme for semilinear wave equations on general bounded polygonal/polyhedral domains, combining an exponential Euler time integrator with a finite element spatial discretization, with initial data $(u^0 ,v^0) \in H^{\gamma}(\Omega)\times H^{\gamma−1} (\Omega)$ for $0 < \gamma \le 1$. In contrast to existing low-regularity error analyses, which are mostly based on Fourier spectral discretizations, our approach applies to general bounded domains and finite element spatial discretizations. We prove rigorous error estimates for low-regularity solutions. The analysis is based on a frequency decomposition of the underlying elliptic operator, used solely as an analytical regularization device and not in the actual implementation, which allows low-regularity techniques to be extended beyond the Fourier spectral framework. The results also indicate that higher-order finite element methods remain advantageous in spatial approximation even for solutions of limited Sobolev regularity. Numerical experiments on different domains and with different polynomial degrees confirm the predicted convergence behavior.