14: p-Multigrid Solver

[philipnickel]

Description

Implement p-multigrid to accelerate the Laplace solve by exploiting the polynomial hierarchy inherent in spectral elements.

Background

p-multigrid uses coarse polynomial spaces (P=1,2,4,...) as preconditioners for the fine space. The Full Approximation Scheme (FAS) extends this to nonlinear problems.

Tasks

• Build polynomial hierarchy (P = 1, 2, 4, 8 or similar)
• Implement restriction operator (fine → coarse)
  • Injection or L² projection
• Implement interpolation operator (coarse → fine)
  • Polynomial interpolation
• Implement smoother
  • Jacobi, Gauss-Seidel, or Richardson
• Implement V-cycle
• Test as solver for Laplace problem
• Compare performance vs direct solve

Acceptance Criteria

• Converges to same solution as direct solve
• Iteration count weakly dependent on P (or independent)
• Faster than direct solve for large problems
• Scalable to high polynomial orders

Technical Notes

• Start with linear multigrid for Laplace
• Can extend to FAS for nonlinear problems if needed
• May be combined with Anderson acceleration