A posteriori error analysis and adaptivity of a space-time finite element method for the wave equation in second order formulation
Provides theoretical guarantees for adaptive space-time methods for wave propagation, benefiting computational scientists solving time-dependent PDEs.
The authors derive rigorous a posteriori error bounds for a space-time finite element method for the wave equation, enabling adaptive mesh refinement. Numerical tests verify the error estimator's convergence and demonstrate a space-time adaptive algorithm.
We establish rigorous \emph{a posteriori} error bounds for a space-time finite element method of arbitrary order discretising linear wave problems in second order formulation. The method combines standard finite elements in space and continuous piecewise polynomials in time with an upwind discontinuous Galerkin-type approximation for the second temporal derivative. The proposed scheme accepts dynamic mesh modification, as required by space-time adaptive algorithms, resulting in a discontinuous temporal discretisation when mesh changes occur. We prove \emph{a posteriori} error bounds in the $L^\infty(L^2)$-norm, using carefully designed temporal and spatial reconstructions; explicit control on the constants (including the spatial and temporal orders of the method) in those error bounds is shown. The convergence behaviour of an error estimator is verified numerically, also taking into account the effect of the mesh change. A space-time adaptive algorithm is proposed and tested numerically.