p-Refinement (Raising Polynomial Order)
Theory: subdivide, or raise the order?
Overview
Mesh convergence usually means smaller elements, but I hear you can raise element order instead. Which is better?
It depends on how smooth the solution is. With h-refinement, error falls only as element size to the power p. p-refinement keeps the mesh and raises shape-function order; for smooth solutions, error falls exponentially with DOF. But where the solution isn't smooth — corners, point loads — raising order barely helps. hp refinement combines both. Let's compare all three at equal DOF in 1D.
Convergence rates
$\| \cdot \|_E$ is the energy norm (error in strain energy), $N$ the DOF, $\lambda$ the singularity strength ($\lambda = \alpha - 1/2$ for a solution $x^{\alpha}$). With a singularity, both h and p versions are capped by $\lambda$.
Hierarchical shape functions
The p-version commonly uses integrated Legendre polynomials. Raising order keeps the lower-order functions and just adds new ones, so stiffness computations can be reused. This example uses them.
The people behind the p-version
The p-version of FEM was developed in the 1970s by Szabó and colleagues at Washington University in St. Louis for structural analysis, and in 1981 Babuška, Szabó and Katz established its convergence theory. The hp theory — exponential convergence for smooth solutions, and with suitable meshes even with singularities — was shown by Babuška and Guo in the 1980s. Szabó's work also led to commercial software for guaranteeing accuracy at stress concentrations.
Worked examples
Example 1: a smooth solution
A 1D problem −u″ = f, u(0) = 0, solution u = sin(πx/2). Relative energy-norm error compared at 16 DOF:
| Method | Relative error |
|---|---|
| h: 16 linear elements | 2.8×10⁻² |
| h: 8 quadratic elements | 1.4×10⁻³ |
| p: 2 elements, order 8 | 6.7×10⁻¹¹ |
Raising order from 1 to 8 on two elements, error went 2.2×10⁻¹ → 2.3×10⁻² → 1.5×10⁻³ → 7.5×10⁻⁵ → 3.0×10⁻⁶ → 9.7×10⁻⁸ → 2.7×10⁻⁹ → 6.7×10⁻¹¹ — more than tenfold per order. With linear elements, doubling DOF only halves the error.
Example 2: a singular solution
Same problem with u = x^0.7. The derivative is infinite at x = 0, like stress near a corner.
| Method | DOF | Relative error |
|---|---|---|
| h: 64 linear elements | 64 | 1.9×10⁻¹ |
| h: 64 quadratic elements | 128 | 1.4×10⁻¹ |
| p: 2 elements, order 8 | 16 | 1.6×10⁻¹ |
| hp: 7 geometric layers, ratio 0.15 | 36 | 3.0×10⁻² |
With a singularity, whether 64 elements or order 8, the error stays above 10 percent.
For x^0.7, λ = 0.2, so uniform refinement cuts error only 13% per halving. Raising order can't make smooth polynomials represent a sharp solution. The hp mesh shrinks elements by 0.15 towards x = 0, keeps the element touching the singularity linear, and raises order outward. Each extra layer multiplied error by about 0.68, reaching 3.0×10⁻² at 36 DOF — about one fifth of uniform quadratics at 128 DOF. Grading small elements around stress concentrations in real structural analysis follows the same idea.
Choosing between them
- Simple geometry and smooth solutions (shell bending, vibration modes) favour p.
- Near corners, cracks, contact edges and material interfaces, grade elements smaller.
- If values keep changing as order rises, suspect a singularity.
- The change as order rises serves directly as an error estimate (p-convergence check).
- High-order elements need high-order geometry (curved boundaries) too, or raising order won't pay off.
“Stress won't settle as order rises”
A designer checking an L-shaped plate in p-version software saw peak stress at the re-entrant corner rise about 10% with every order increase from 4 to 8. In smooth regions p-results settle within a few orders, so failing to settle itself signals a singularity. The drawing showed a 2 mm fillet at the corner; modelling it made orders 6 and 8 differ by under 0.5%, giving values usable for fatigue assessment.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Relying on p alone at singularities | Error stalls | Combine with geometric grading |
| Linear geometry approximation | High order wasted | Use curved geometry |
| High order on distorted elements | Accuracy drops | Improve element shape |
| Excessive order | Ill-conditioning | Use hierarchical functions |
| Judging by local stress only | Misread convergence | Check energy too |
I'd like to learn related topics.
See shape functions by order in the 1D shape function tool. Related pages include h-refinement, adaptive mesh refinement, singularities and meshing, Richardson extrapolation and the mesh convergence tool.
Related Topics
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