An ultraweak space-time variational formulation for the wave equation: Analysis and efficient numerical solution

We introduce an ultraweak space-time variational formulation for the wave equation, prove its well-posedness (even in case of minimal regularity) and optimal inf-sup stability. Then, we a tensor product-style Petrov–Galerkin discretization with discrete stability, obtained by non-standard definition trial space. As consequence, numerical approximation error is equal to residual, which particularly useful posteriori estimation. For arising linear systems space time, efficient solvers that appropriately exploit equation structure, either at preconditioning level or phase using tailored Galerkin projection. This method shows competitive behavior concerning wall-clock accuracy memory as compared standard time-stepping particular low regularity cases. Numerical experiments 3D (in space) illustrate our findings.