2026/07/17 by Jiachuan Cao, Benjamin Dörich, Buyang Li
#math.NA #cs.NA
We study nonlinear wave equations with dynamic boundary conditions on smooth bounded domains and analyze a fully discrete approximation in the low-regularity regime. The method combines isoparametric bulk--surface finite elements of degree k with an exponential integrator in time. Assuming only bounded energy of the exact solution, we prove convergence of the displacement--velocity pair in the weak norm L2(Ω;Γ)× H-1(Ω;Γ). The scheme achieves first-order convergence in time and spatial convergence of order h2/3 for k=1 and h(k+2)/(k+3) for k≥ 2. In particular, these rates show that higher-order finite elements retain a provable asymptotic advantage even at low regularity. A central difficulty is that the continuous and discrete bulk--surface problems are posed on different geometries and must therefore be compared directly in weak norms. To address this, we develop a weak-norm framework for non-conforming geometries based on lift and adjoint-lift operators, combined with a frequency-decomposition argument. To the best of our knowledge, this is the first fully discrete low-regularity convergence result for nonlinear wave equations with dynamic boundary conditions in a non-conforming bulk--surface finite element setting. Numerical experiments confirm the predicted rates and illustrate the improved efficiency of higher-order methods.