A study of some multigrid ideas

摘要

In an effort to understand certain ideas and concepts associated with multigrid iterations, we give an in-depth study of a particular simple problem. We consider a standard finite-difference system associated with the two-point boundary-value problem -(pu′)′ + bu′ + qu = 0, u(0) = u(1) = 0. The operators I2hh, Ih2h are “operator” based interpolation and projection operators, while the smoothers are the damped Jacobi iterations with parameter a > 0. We determine the exact rates of convergence for the “two-grid” scheme and upper bounds (<1) for the multigrid schemes. Experimental results are discussed.