## Two-Grid hp Solver

The hp self-adaptive strategy described in the previous chapters employs a fine grid solution to guide optimal refinements. Hence, to efficiently solve the problem of interest over the fine grid becomes essential in order to apply this technology to real-world applications. We note that the fine grid used in the described self-adaptive strategy contains roughly 8 ( p+2)

( p+1)3 more unknowns than the corresponding coarse grid, where p is the polynomial order of approximation. For example, for p = 2, the fine grid contains approximately 27 times more unknowns than the corresponding coarse grid. This implies that, for computing the fine grid solution, we need almost three orders of magnitude more CPU time than for computing the coarse grid solution, if a direct solver is employed.