4 Tips for Implementing Multigrid Methods on Domains with Holes

1. 4 Tips for Implementing Multigrid Methods on Domains with Holes Scott Hawley & Richard Matzner Center for Relativity University of Texas at Austin APS April Meeting, 2003

2. Multigrid for Constrained Ev. • Want to do ‘constrained evolution’ • with excision • Using 2nd order finite difference methods • Need a fast elliptic solver • Try (vertex-centered) multigrid

3. Sad News: You Won’t Get 2nd O. • We were advised that obtaining 2nd order convergence near holes is impossible or impractical, especially in 3D,… • Unless we… • use ‘fancy’ methods (e.g. deferred correction) • OR use a cubical excision region, having same domain on all multigrid levels

4. Our Stubbornness • Our application uses Kerr-Schild type data • Analytic, exact inner BC (f=1) • “Should be a simple implementation…” • Didn’t want to be limited to excising only cubes

5. Our Test Case • Try solving nonlinear Poisson eq: D2 u(x,y,z) +u2(x,y,z) = f(x,y,z) • Inner BC: Dirichlet, using exact value u(x,y,z) • Outer BC: Dirichlet or Robin: [r(u-1)],r=0 • SUCCESS! Get global 2nd O conv.

6. Tip 1: Boundary/Restriction Scheme • Dirichlet BC on edge of hole (circled X’s) using u(x,y,z) • Use weighted restriction if no neighboring fine grid points are on excision mask, otherwise just “copy” • To anyone: Why does this work? FYI, NOT:

7. Tip 2: Focus Smoothing Where Needed Error concentrated near holes: Define “Extra Smoothing Region”: Smooth ESR points twice as much as normal points

8. Results w/ Extra Smoothing Region

9. Pre- vs. Post-CGC Smoothing Two Holes • Tip 3: Pre-CGC smooths more effective than Post-CGC smooths Note: For multiple V-cycles, you need Post-CGC smooths!

10. Tip 4: ‘Fake Robin’ Outer BC is OK… • Robin: [r(u-1)],r = 0 • Can just use simple, first order deriv in direction normal to faces of cubical domain [Alcubierre]

11. Wrapping Up • 4 Tips: • Put boundary points at edge of hole, but don’t use them in weighted restriction • Use Extra Smoothing Region • Try more Pre- than Post-CGC smooths • For Robin BC: Use first-order perp. deriv • Future Work: • Black hole solutions! • Need to extrapolate for inner BC? • Parallelization