RELACS Projects: Multigrid and Multiresolution Solvers

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Projects

• Hyperbolic MG

  This is connected to Mark's work on MG for MHD. They use a predictor-corrector formalism. The main idea is that high frequencies introduced by interpolation first need to be damped out, and then the corrector restores accuracy. This is exactly what Jameson does for the steady state. However, we still have the problem of inaccuracy for the coarse operator (phase error). Can this be corrected using FAS?

  Construct a Helmholtz sovler testbed in PETSc

  Construct an advection sovler testbed in PETSc

  A big question for me is to disentangle the separate things that we might want out of the solver, and what impact the different methods have for each:

  • Stability
  • Accuracy
  • Conservation
  For example, in our test which an advecting block. The implicit Backwards-Euler method is stable, and globally conservative, but it smears out the block. This is what is meant by the adjective "diffusive". A higher order method would lessen this effect, and people like Germond would argue that the method is conservative, over the support of each basis function rather than over a cell. However, is this good enough when other processes might be nonlinearly coupled to the advected quantity, as in magma dynamics?

• Nonlinear AMG

  We can use the equivalent of DEIM with AMG coarse vectors instead of POD vectors to construct an interpolant for the nonlinear part. The only problem I forsee is the number of evaluations of F. There are a very small number of POD vectors, but AMG can have many. We might need further simplification.

  • -ksp_type cg
  • -pc_type gamg
  • -pc_gamg_agg_nsmooths 1
  • -pc_gamg_threshold 0.02 # [0 - 0.1]
  • #-mg_levels_ksp_type richardson
  • -mg_levels_ksp_type chebyshev
  • -mg_levels_pc_type sor
  • #-mg_levels_pc_type jacobi
  • -mg_levels_ksp_max_it 2 # [1-8]
• Euler stability

  T. Tao's diagram for Euler stability (search in blog for Euler). I think we could code up his proof diagram and look for solutions, etc.

• Distributive smoothers for nonlinear Stokes

Finished Projects

• Segmental Refinement Multigrid (SRMG)

  Motivation

  Brandt proposed an \(\mathcal{O}(\log N)\) memory method for computing the solution to elliptic equations called segmental refinement multigrid. It solves individual domains of the fine problem completely independently and only communicated through the coarse problem. This relies on the fact that high frequency perturbations in the boundary conditions for elliptic equations die out exponentially into the interior.

  Background

  There are only two papers about SR, one by Diskin et.al. and one in SISC by Adams et.al.

  There is a beautiful SIAM Review article on eigenfunctions of the Laplacian which has good citations on the central potential theory result about decay of boundary perturbations (interior regularity).

  Project

  1. I have had an idea for an even simpler way to look at interior regularity. I will list the steps:
     1. Setup multigrid (structured ex5 or unstructured ex12) for the Laplacian
     2. Evaluate the error on each grid and plot
     3. Look at log-log graph for discretization error and get constant C
     4. Now we solve the same problem, but the boundary condition comes from the coarsest level. We can use either the projection of the exact coarse function or its interpolant
     5. Evaluate the error on each grid and plot
     6. Shrink square away from the boundary until we are at discreization error
     7. If this middle square tends to a finite limit as we increase resolution, then SR truly works and Mark has a problem in his code.
     8. We should be able to prove this, one way or the other, using the theory from Nirenberg.
     Spectral solution
     1. Setup spectral solver on the square. This means get the Green function with homogeneous Dirichlet conditions (Jackson, p.89) and do the first integral in Green's Theorem, and then superpose this by the expansion solution to Laplace with the inhomogeneous boundary conditions.
     2. Give an exact complexity for the solution, including the quadrature.
     3. Evaluate the solution on a patch made by cutting the domain in two parts with a line in y. The Green function changes in a predictable way. The boundary condition is given by the spectral solution. Show that the error is still at discretization error for this modification.
     5. Double the spectral resolution for the patch
     6. Plot the patch error and show that it decays away from the boundary exponentially
     7. Show that the error decay exponentially by examining the sinh term in the solution, and assuming that only frequencies greater than k exist in the error for the boundary condition.
     8. Repeat this process, moving the domain cut into the region where the error is once again below discretization error.
     9. Prove that the total distance the domain cut moves is bounded independent of the number of refinements. What is the weakest condition on the decay of the error which gives a uniform bound?
  2. Exact solve on a coarse (transition) grid, both a spectral solution and a direct FE solution. Then set up two partitions with the calculated overlap, and run with "full buffer protocol". We need to evalaute the error from the MS of course, but I think we should evaluate the "interior regularity" by looking at the Fourier power spectrum of the error as a function of the wavenumber of the boundary perturbation. Exact solves on every level will give us the answer to our questio nabout error decay into the interior.
  3. Setup the Laplace equation, and look at the effect of high frequency perturbations of both the boundary values and the boundary itself. Compare the error in the solution, and use Fourier Transform to look at its components in frequency space.
  4. Setup a toy SR where everything is sequential so that its easy to have complete information.
  5. Veryify in toy SR that low freq. error decay at the correct rate by using MMS.