Influence of (pointwise) Gauss-Seidel relaxation on the error Poisson equation, uniform grid Error of initial guess Error after 5 relaxation Error after 10 relaxations Error after 15 relaxations

LU=F h 2h 4h L h U h =F h L 2h U 2h =F 2h L 4h U 4h =F 4h

Local relaxation approximation smooth

h2h Local relaxation approximation smooth

Coarse Grid Correction Approximate solution Error Residual equation, where 1. Fine grid relaxation Goto 1 2. Coarse grid eq. 3. h old h new uu h2 v ~~~  Full Approximatioin Scheme (FAS): defect correction

interpolation (order l+p) to a new grid interpolation (order m) of corrections relaxation sweeps algebraic error < truncation error residual transfer enough sweeps or direct solver * *** Full MultiGrid (FMG) algorithm... * h0h0 h 0 /2 h 0 /4 2h h

F cycle h0h0 h 0 /2 h 0 /4 2h h ***... * interpolation (order l+p) to a new grid interpolation (order m) of corrections relaxation sweeps algebraic error < truncation error residual transfer enough sweeps or direct solver * residual transfer no relaxation