1. Geometric (Classical) MultiGrid

2. Linear scalar elliptic PDE (Brandt ~1971)  1 dimension Poisson equation  Discretize the continuum x0x0 x1x1 x2x2 xixi x N-1 xNxN x=0x=1 Grid: Let local averaging

3. Linear scalar elliptic PDE  1 dimension Laplace equation  Second order finite difference approximation => Solve a linear system of equations Not directly, but iteratively => Use Gauss Seidel pointwise relaxation

4. 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

5. The basic observations of ML  Just a few relaxation sweeps are needed to converge the highly oscillatory components of the error => the error is smooth  Can be well expressed by less variables  Use a coarser level (by choosing every other line) for the residual equation  Smooth component on a finer level becomes more oscillatory on a coarser level => solve recursively  The solution is interpolated and added

6. TWO GRID CYCLE Approximate solution: Fine grid equation: 2. Coarse grid equation: h2 v ~~~ h old h new uu  Residual equation: Smooth error: 1. Relaxation residual: h2 v ~ Approximate solution: 3. Coarse grid correction: 4. Relaxation

8. TWO GRID CYCLE Approximate solution: Fine grid equation: 2. Coarse grid equation: h old h new uu h2 v ~~~  Residual equation: Smooth error: 1. Relaxation residual: h2 v ~ Approximate solution: 3. Coarse grid correction: 4. Relaxation 1 2 3 4 5 6 by recursion MULTI-GRID CYCLE Correction Scheme

9. interpolation (order m) of corrections relaxation sweeps residual transfer enough sweeps or direct solver *... * h0h0 h 0 /2 h 0 /4 2h h V-cycle: V     

10. Hierarchy of graphs Apply grids in all scales: 2x2, 4x4, …, n 1/2 xn 1/2 Coarsening Interpolate and relax Solve the large systems of equations by multigrid! G1G1 G2G2 G3G3 GlGl G1G1 G2G2 G3G3 GlGl

11. Linear (2 nd order) interpolation in 1D x1x1 x2x2 x F(x)

12. i S(i) (U lb,V lb ) (U rt,V rt )(U lt,V lt ) (U rb,V rb ) (x 2,y 2 )(x 1,y 2 ) (x 2,y 1 )(x 1,y 1 ) (x 0,y 0 ) Bilinear interpolation C(S(i))={rb,rt,lb,lt}

13. i S(i) (U lb,V lb ) (U rt,V rt )(U lt,V lt ) (U rb,V rb ) (x 2,y 2 )(x 1,y 2 ) (x 2,y 1 )(x 1,y 1 ) (x 0,y 0 ) (U l,V l )(U r,V r )

14. From (x,y) to (U,V) by bilinear intepolation

15. The fine and coarse Lagrangians For each square k add an equi-density constraint eqd(k) = current area + fluxes of in/out areas – allowed area = 0 is the bilinear interpolation from grid 2h to grid h At the end of the V-cycle interpolate back to (x,y)