4: Convergence Study and Validation Plots

[philipnickel]

Description

Conduct systematic convergence studies demonstrating the spectral accuracy of the solver. Produce publication-quality plots showing h-convergence and p-convergence.

Background

Spectral element methods should exhibit:

• p-convergence: Exponential error decay with polynomial order (for smooth solutions)
• h-convergence: Algebraic error decay O(h^P) with mesh refinement

Tasks

• Implement error computation (L² norm against analytical)
• p-convergence study
  • Fix mesh (e.g., 4 elements)
  • Vary P = 2, 3, 4, 5, 6, 7, 8, 10, 12
  • Record error for each P
• h-convergence study
  • Fix P (e.g., P = 4 and P = 6)
  • Vary N_el = 2, 4, 8, 16, 32
  • Record error for each mesh
• Generate plots
  • p-convergence: error vs P (semi-log scale)
  • h-convergence: error vs h (log-log scale)
  • Cost-efficiency: error vs DOF (log-log scale)

Acceptance Criteria

• p-convergence plot shows straight line on semi-log (exponential convergence)
• h-convergence plot shows slope ≈ P on log-log (algebraic O(h^P))
• Results consistent between interior Φ error and surface w̃ error
• Plots are publication quality (labeled axes, legend, clean formatting)

Deliverables

• p_convergence.pdf
• h_convergence.pdf
• cost_efficiency.pdf