Robust solutions to PDEs with multiple grids

In this paper we will discuss some approaches to fault-tolerance for solving partial differential equations. In particular we will discuss how one can combine the solution from multiple grids using ideas related to the sparse grid combination technique and multivariate extrapolation. By utilising the redundancy between the solutions on different grids we will demonstrate how this approach can be adapted for fault-tolerance. Much of this will be achieved by assuming error expansions and examining the extrapolation of these when various solutions from different grids are combined