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Uniform-degree polygon corner spaces have cubic degree-of-freedom growth

Statement

For $k \geq 1$ , let a matching triangulation $T_{k}$ of a polygon have at most $C_{0} k$ triangles. Assign a uniform total polynomial degree $p_{k} \geq 1$ with $p_{k} \leq C_{1} k$ . The algebraic space of continuous piecewise $P_{p_{k}}$ functions has dimension

N_{k} \leq C_{2} k^{3},

where $C_{2}$ depends only on $C_{0}$ and $C_{1}$ . The same upper bound holds for any subspace imposing homogeneous boundary traces. This is an upper count only; it asserts no hp approximation rate.

Facts & Assumptions

Given: Integer $k \geq 1$ , a matching triangulation $T_{k}$ with $# T_{k} \leq C_{0} k$ , and uniform degree $1 \leq p_{k} \leq C_{1} k$ .

[F1]

Variable-degree conforming hp spaces on matching affine meshes: the continuous polynomial space is a linear subspace of the broken product of its cell polynomial spaces.

Proof

1.1F1givenalgebra

On a triangle, monomials $x^{i} y^{j}$ with $i, j \geq 0$ and $i + j \leq p_{k}$ form a basis of $P_{p_{k}}$ . Counting by total degree gives $\dim P_{p_{k}} = \sum_{r = 0}^{p_{k}} (r + 1) = (p_{k} + 1) (p_{k} + 2) / 2$ .

2.1F1step 1.1givenalgebra∎

The broken product over $T_{k}$ therefore has dimension $# T_{k} (p_{k} + 1) (p_{k} + 2) / 2$ . By [F1], trace matching and any zero-boundary conditions define a linear subspace, whose dimension cannot exceed the broken dimension. Using $# T_{k} \leq C_{0} k$ and $p_{k} \leq C_{1} k$ , and $k \geq 1$ , gives $N_{k} \leq C_{0} k (C_{1} k + 1) (C_{1} k + 2) / 2 \leq C_{2} k^{3}$ for an explicit finite $C_{2}$ depending only on $C_{0}, C_{1}$ .

Depends on

Variable-degree conforming hp spaces on matching affine meshes

Used by

Exponential hp energy convergence for weighted-analytic polygon corners Theorem

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Yanchen He, Paul Houston, Christoph Schwab, and Thomas P. Wihler, Exponential Convergence of hp-ILGFEM, version 1 (standard reference, not scraped)
Leszek F. Demkowicz, Mathematical Theory of Finite Elements (standard reference, not scraped)