Vertex / edge
Vertex modes use only ℓ₀, ℓ₁. Edge modes put one factor at order k ≥ 2 and pin the transverse directions to a vertex factor so continuity lives on that edge alone.
Raise p, keep the mesh
Hierarchical Lobatto shape functions are the workhorse of the p-version. The viewer below evaluates the same 1-D library as a typical high-order code, then builds tensor-product modes in 1-D, 2-D, and 3-D — contours, warped surfaces, volume slices, and isosurfaces you can tune live.
φ = —
min · max
h-version versus p-version
h-version
Keep the polynomial degree fixed (usually p = 1 or 2) and refine the mesh size h. Algebraic rates follow: ‖u − uh‖L² = O(hp+1) and ‖∇(u − uh)‖ = O(hp) when the solution is smooth enough.
p-version
Keep the mesh, raise the degree. For analytic solutions the error decays exponentially in p. Near singularities an hp strategy — geometric refinement plus high order away from the tip — recovers exponential rates again.
The plots below show the classical algebraic rates on a sequence of meshes for p = 1…7. Steeper slopes at higher p are exactly the O(hp+1) and O(hp) predictions for the field and its gradient.
Convergence. Relative L² error in u (left) falls as O(hp+1); the gradient (right) as O(hp).
Why Lobatto, not monomials
A naïve basis {1, ξ, ξ², …, ξp} on [−1, 1] becomes severely ill-conditioned as p grows. Integrated Legendre (Lobatto) polynomials vanish at the endpoints for k ≥ 2 and stay nearly orthogonal in the energy inner product, so hierarchical element matrices remain usable at high p.
ℓ₀(ξ) = (1 − ξ)/2
ℓ₁(ξ) = (1 + ξ)/2
ℓ₂(ξ) = ½√(3/2) (ξ² − 1)
ℓ₃(ξ) = ½√(5/2) (ξ² − 1) ξ
ℓ₄(ξ) = ⅛√(7/2) (ξ² − 1)(5ξ² − 1)
ℓk(ξ) = ck (ξ² − 1) P′k−1(ξ) for k ≥ 2
ℓ₁(ξ) = (1 + ξ)/2
ℓ₂(ξ) = ½√(3/2) (ξ² − 1)
ℓ₃(ξ) = ½√(5/2) (ξ² − 1) ξ
ℓ₄(ξ) = ⅛√(7/2) (ξ² − 1)(5ξ² − 1)
ℓk(ξ) = ck (ξ² − 1) P′k−1(ξ) for k ≥ 2
Families on the reference element
Face (3D)
On a hexahedron a face mode has two high-order factors in the face plane and one vertex factor normal to the face — zero on the other five faces.
Bubble
All free directions use k ≥ 2, so the mode vanishes on the entire boundary and can be statically condensed.
1D: φ = ℓᵢ(ξ)
2D: φ = ℓᵢ(ξ) ℓⱼ(η)
3D: φ = ℓᵢ(ξ) ℓⱼ(η) ℓₖ(ζ)
2D: φ = ℓᵢ(ξ) ℓⱼ(η)
3D: φ = ℓᵢ(ξ) ℓⱼ(η) ℓₖ(ζ)
Reference bubble set
Diagonal bubbles ℓk(ξ) ℓk(η) for k = 2…7 from the original ParaView export — reproduce them in the viewer with dimension 2D, family Bubble, and i = j = k.
Interior bubbles. Blue / red opposite signs; green near zero. Every panel vanishes on all four edges.
Practical notes
- Quadrature. Integrate at least degree 2p − 2, or expected rates disappear.
- Condensation. Bubbles (and often faces) have local support and can be eliminated before assembly.
- Singularities. Pure p on a coarse mesh stalls at tips; use hp geometric refinement.
- Spectral cousins. GLL nodal bases use the same orthogonal polynomials, but are nodal rather than hierarchical.
Contours above match Šolín, Segeth & Doležel, Higher-Order Finite Element Methods, and the MATLAB Lobatto library used to write shFuncs.vtu.