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The HDG Boundary Layer case... involves a heat source being applied to a vortex flow around a circular annulus of radius r = [ 1 , 2 ]. The numerical heat flux at the inner boundary is checked with the analytical heat flux at the inner boundary to a specified tolerance. All boundaries are set to a thermal condition which is a function of r and θ. The applied heat source is also a function of r and θ, and the initial condition is set to 1.

• Equation Set: Scalar 2D
• Nonlinear Solver: Newton
• Basis Type: Hierarchical
• Initial/Max CFL: 10³⁰
• Target residual: 10^-11
• Preconditioning:
  • Type: ILU (Left)
  • Reordering: MDF
  • Outer GMRES iterations (# restarts): 20
  • Inner GMRES iterations (# Krylov vectors): 200

Tested Outputs

• L₂ error (compared to an exact solution function)
• Numerical heat flux on inner boundary
• Exact heat flux error on inner boundary
  
• L_∞ of the exact heat flux error

• Vortex Flow
  
  
• Boundary Condition:
  
• Heat Source:
  
  where
  
• Non-dimensional viscosity:
  

• Solution Order: 2
• Iterations to solve: 2
• All outputs tested to a tolerance of 10^-12

Job File:
Scalar2d_CircleBoundaryLayer

• Solution Order = 1
• Additional tested outputs: Heat flux adjoint residual

• Adaptation parameters:
  • Adaptation Iterations: 2
  • Adaptation Method: Fixed fraction
  • Output Adapted: Heat flux
  • Anisotropy Method: Hessian

• Iterations to solve on each mesh: 2
• All outputs tested to a tolerance of 10^-10

Job File:
Scalar2d_CircleBoundaryLayer_Adapt