A posteriori error estimates for non-linear parabolic equations

We consider space-time discretizations of non-linear parabolic equations. The temporal discretizations in particular cover the implicit Euler scheme and the mid-point rule. For linear equations they correspond to the well-known A-stable θ-schemes. The spatial discretizations consist of standard conforming finite element spaces that can vary from one time-level to the other. The spatial meshes may be locally refined, but must be isotropic. For these discretizations we derive a residual a posteriori error estimator which yields upper and lower bounds on the error. The ratio of upper and lower bounds does not depend on any mesh-size in space or time nor on any relation between both. In particular there is no restriction on the relative size of the temporal and spatial mesh-sizes.