In Sects. 4 and 5 the approach will be extended to more general nonlinear evolutionary inequalities.
In the following, we introduce the considered full discretisations for nonlinear evolutionary inequalities.
The convergence of full discretisations by implicit Runge Kutta and nonconforming Galerkin methods applied to nonlinear evolutionary inequalities is studied.
Moreover, it extends the analysis of stiffly accurate Runge Kutta time discretisations for nonlinear evolution equations given in [ 15] to nonlinear evolutionary inequalities.
However, to our knowledge, the present work is the first contribution where implicit Runge Kutta time discretisations are studied for nonlinear evolutionary inequalities.
To our knowledge, this work is the first contribution, where a class of implicit Runge Kutta methods is investigated for nonlinear evolutionary inequalities.
The present work generalises [ 8], where full discretisations based on the implicit Euler method and low-order finite element approximations applied to nonlinear evolutionary inequalities are studied.
In this section, we introduce a simple though characteristic model problem, which shall serve as an illustration of the general framework of nonlinear evolutionary inequalities.
In regard to abstract evolutionary inequalities treated in Sect. 4, the first term in the variational inequality (2.2) corresponds to the duality pairing of ∂ t u(⋅, t)− f(⋅, t)∈ X∗ and v(⋅, t)− u(⋅, t)∈ X.
Under hypotheses close to the existence theory of nonlinear evolutionary inequalities, dispensing with unnatural regularity requirements on the exact solution to the problem, we are able to establish a convergence result for the piecewise constant in time interpolant.
However, as in the present work our focus is on the derivation of a convergence result for stiffly implicit Runge Kutta methods applied to evolutionary inequalities, but not on their rate of convergence, we do not further exploit this point.
