Vector & Higher-Rank Operators
Convergence for operators that produce tensor output or act on vector fields: hessian, jacobian, divergence, curl. The test vector field is u(x) = [sin(πx₁) + 0.5cos(2πx₂), exp(x₁x₂)] on [0,1]² with all components having nonzero partials.
"Which PHS order?" plots below show PHS3/5/7 at fixed poly_deg = 3. Holding the polynomial augmentation constant isolates the PHS-order effect: curves should be parallel on log-log, differing only in the error constant.
IMQ and Gaussian h-refinement is covered on its own page (Shape-Parameter Bases) — shape-parameter behavior is dominated by the ε-vs-spacing interaction and is discussed there for a representative subset of operators.
Hessian
The Hessian assembles second partials ∂²u/∂xᵢ∂xⱼ into an N × D × D tensor. Error is reported as a Frobenius NRMSE. Because one of the four entries is a mixed partial, the Hessian inherits the mixed-partial caveat — the non-converging combos are the same as for the mixed partial operator.
Excluded combinations
Same PHS set as mixed partial: PHS1/p=1, PHS3/p=1, PHS3/p=2, PHS5/p=2 are omitted. Use PHS5/p=3 or higher for the full Hessian. Shape-parameter bases: see Shape-Parameter Bases.
Which PHS order?
At poly_deg = 3, all three PHS orders converge at O(h²) (the second-derivative rate). PHS7 has the smallest error constant.
How much polynomial degree?
Jacobian
The Jacobian of a vector field is an N × Dᵢₙ × Dₒᵤₜ tensor of first partials — no second derivatives, no mixed-partial issue. Convergence matches the gradient cleanly.
Which PHS order?
Parallel O(h³) curves across PHS3/5/7.
How much polynomial degree?
Divergence (∇·)
∇·u = ∂u₁/∂x₁ + ∂u₂/∂x₂ — a sum of first partials. Convergence rates match the gradient; all PHS orders at matched poly_deg are well-behaved.
Which PHS order?
Parallel O(h³) curves across PHS3/5/7.
How much polynomial degree?
Curl (∇×, 2D)
In 2D, ∇×u = ∂u₂/∂x₁ − ∂u₁/∂x₂ (the scalar z-component). Like divergence, this is a first-derivative operator and follows the same convergence profile.
Which PHS order?
Parallel O(h³) curves across PHS3/5/7.
How much polynomial degree?
