Issues related to space–time adaptivity for a class of nonlinear and time-dependent problems are discussed. The dG(k)-methods are adopted for the time integration, and the a posteriori error control is based on the appropriate dual problem in space–time. One key ingredient is to decouple the error generation in space and time with a hierarchical decomposition of the discrete space of dual solutions. The main idea put forward in the paper is to increase the computational efficiency of the adaptive scheme by avoiding recursive adaptations of the whole time-mesh; rather, the space-mesh and the time-step defining each finite space–time slab are defined in a truly sequential fashion. The proposed adaptive strategy is applied to the coupled consolidation problem in geomechanics involving large deformations. Its performance is investigated with the aid of a numerical example in 2D. 1. Compute z h | S n ( k ) based on z h ( t n − 1 − ) that was computed for the previous time slab S n − 1 . Then, solve for the enhanced dual solution z ̃ ∗ | S n ( k ) from the decoupled dual problem A n ∗ ( z h ( k ) ; δ z ∗ , z ̃ ∗ ( k ) ) = L n ∗ ( z h ( k ) ; δ z ∗ , z ̃ H ∗ ( t n + ) ) , whereby it is noted that z ̃ ∗ ( t n + ) has been replaced by the background dual solution z ̃ H ∗ ( t n + ) as the load (or data) for the current space–time slab. Compute the error contributions E n , SOL ( k ) , E ( s ) n , FEM ( k ) , etc. Check the stopping criterion in (68): If E n ( k ) ≤ TOL Δ t n ( k ) T then exit and take a new time step. Refine the space-mesh, the time interval or both: If α − E ( t ) n , FEM ( k ) ≤ E ( s ) n , FEM ( k ) ≤ α + E ( t ) n , FEM ( k ) then refine in space and time uniformly. Else if α + E ( t ) n , FEM ( k ) ≤ E ( s ) n , FEM ( k ) then refine in space: M h , n ( k ) → M h , n ( k + 1 ) , I n ( k ) = I n ( k + 1 ) Else if E ( s ) n , FEM ( k ) ≤ α − E ( t ) n , FEM ( k ) then refine in time: I n ( k ) → I n ( k + 1 ) , M h , n ( k ) = M h , n ( k + 1 ) 
A sequential-adaptive strategy in space–time with application to consolidation of porous media