Doubly nonlinear evolution equations of second order: Existence and fully discrete approximation

Existence of solutions for a class of doubly nonlinear evolution equations of second order is proven by studying a full discretisation. The discretisation combines a time stepping on a nonuniform time grid, which generalises the well-known Störmer–Verlet scheme, with an internal approximation scheme. The linear operator acting on the zero-order term is supposed to induce an inner product, whereas the nonlinear time-dependent operator acting on the first-order time derivative is assumed to be hemicontinuous, monotone and coercive (up to some additive shift), and to fulfill a certain growth condition. The analysis also extends to the case of additional nonlinear perturbations of both the operators, provided the perturbations satisfy a certain growth and a local Hölder-type continuity condition. A priori estimates are then derived in abstract fractional Sobolev spaces. Convergence in a weak sense is shown for piecewise polynomial prolongations in time of the fully discrete solutions under suitable requirements on the sequence of time grids.