Conforming isogeometric Galerkin spaces and h/p/k refinement

For one admissible patch map $F:\widehat{\mathrm{\Omega }}\to \mathrm{\Omega }$, let $S_h$ be a finite tensor-product B-spline space on a knot grid refining the geometry grid, and let $W$ be the positive denominator of the geometry representation. The parametric rational analysis space is ${\widehat{V}}_h:=\{s\mathrm{/}W:s\in S_h\}\mathrm{.}$ Its physical push-forward is $\{\widehat{v}\circ F^{-1}:\widehat{v}\in {\widehat{V}}_h\}$. The homogeneous-Dirichlet conforming space is the subspace of these functions that belongs to $H_0^1(\mathrm{\Omega })$; the unconstrained conforming version is the subspace belonging to $H^1(\mathrm{\Omega })$. These spaces are finite dimensional because the tensor B-spline coefficient set is finite.

For vector-valued elasticity in ${\mathbb{R}}^d$, use the componentwise product ${\widehat{V}}_h^d:=({\widehat{V}}_h{)}^d$, push it forward component by component, and intersect with $H^1(\mathrm{\Omega };{\mathbb{R}}^d)$ or $H_0^1(\mathrm{\Omega };{\mathbb{R}}^d)$ as appropriate. This remains finite dimensional because it is a subspace of the finite product of scalar coefficient spaces. Trace equality is required componentwise on each shared full face.

For finitely many matching patches, use one such local rational space on each patch. A global trial function is a piecewise physical function whose traces agree pointwise on every shared full face and which belongs to the chosen global $H^1$ or $H_0^1$ space. Under Countable Choice, the vector-valued trace-matching lemma Matching NURBS patch traces give global H1 conformity proves the H1 part of this interface rule componentwise, including the Cartesian product used for elasticity. Nonmatching interfaces are excluded; a mortar or Nitsche method requires a separate stability and consistency result.

The refinement labels have the following meanings.

• h-refinement inserts a new knot into a nonempty span at fixed degree and updates homogeneous control data by Single new-knot insertion for a B-spline curve. Theorem Homogeneous knot insertion preserves an admissible NURBS patch proves exact map preservation for this operation.
• p-refinement raises a coordinate degree and replaces the homogeneous representation only when an exact representation with positive new weights is supplied. A degree label alone does not certify geometry preservation or an approximation rate.
• k-refinement raises degree and chooses knot multiplicities to obtain a declared continuity, commonly the maximal $C^{p_r-1}$ continuity at simple interior knots. Exact geometry preservation requires the same positive homogeneous representation check as p-refinement.

On each cell, the pullback of scalar gradients and of the linear-elasticity strain is computed using Bi-Lipschitz NURBS maps preserve H1 and energy norms. Exact geometry does not assert exact quadrature. A discretization using an approximate map or quadrature must retain its separate form and load defects.