A space–time formulation and improved spatial reconstruction for the “divide-and-conquer” multiscale method

A time-discontinuous space–time formulation for the “divide-and-conquer” multiscale method is presented. This multiscale method aims at stable and accurate solutions of transient convection–diffusion-reaction problems. The method is constituted by a time-discontinuous space–time finite element method on the fine-scale level, an explicit procedure for advancing the coarse-scale solution on the respective level, and inter-scale operators providing the data transfer between the coarse- and fine-scale level in both directions. The results from three numerical test cases show that for both problematic flow regimes, that is, the regime of dominant convection and the regime of dominant convection and absorption, the proposed method provides completely stable solutions, which are not achieved by standard stabilized methods, particularly for the later regime. For remedying a still to be noted shortcoming of the proposed method, that is, a too smooth resolution of sharp-gradient regions in the solution field, considerable improvement can be provided by the use of cubic spline interpolation as the spatial reconstruction (i.e., the spatial coarse-to-fine inter-scale) operator.