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

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.