Mandel, JanSousedık, Bedrich2021-11-182021-11-182010http://hdl.handle.net/11603/23371The 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 Widlund [2008]. Also, space limitation does not allow us to even the 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-hand-side at any other point. Both multigrid and domain decomposition combine a global correction in 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.8 pagesen-USThis item is likely protected under Title 17 of the U.S. Copyright Law. Unless on a Creative Commons license, for uses protected by Copyright Law, contact the copyright holder or the author.Text