Coarse Spaces over the Ages

The objective of this paper is to explain the principles of the design of a coarse space in a simplified way and by pictures. The focus is on ideas rather than on a more historically complete presentation. That can be found, e.g., in [ 28]. Also, space limitation does not allow us even to mention many important methods and papers that should be rightfully included. The coarse space facilitates a global exchange of information in multigrid and domain decomposition methods for elliptic problems. This exchange is necessary, because the solution is non-local: its value at any point depends on the right-handside at any other point. Both multigrid and domain decomposition combine a global correction in the coarse space with local corrections, called smoothing in multigrid and subdomain solves in domain decomposition. In multigrid the coarse space is large (typically, the mesh ratio is 2 or 3 at most) and the local solvers are not very powerful (usually, relaxation). In domain decomposition, the coarse space is small (just one or a few degrees of freedom per subdomain), and the local solvers are powerful (direct solvers on subdomain). But the mathematics is more or less the same.