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# PETSCSPACEPTRIMMED

“ptrimmed” - A `PetscSpace` object that encapsulates a trimmed polynomial space. Trimmed polynomial spaces are defined for \(k\)-forms, and are defined by \( \mathcal{P}^-_r \Lambda^k(\mathbb{R}^n) = mathcal{P}_{r-1} \Lambda^k(\mathbb{R}^n) \oplus \kappa [\mathcal{H}_{r-1} \Lambda^{k+1}(\mathbb{R}^n)], \) where \(\mathcal{H}_{r-1}\) are homogeneous polynomials and \(\kappa\) is the Koszul differential. This decomposition is detailed in ``Finite element exterior calculus’’, Arnold, 2018.

## Notes

## Trimmed polynomial spaces correspond to several common conformal approximation spaces in the de Rham complex

In \(H^1\) (\(\sim k=0\)), trimmed polynomial spaces are identical to the standard polynomial spaces, \(\mathcal{P}_r^- \sim P_r\).

In \(H(\text{curl})\), (\(\sim k=1\)), trimmed polynomial spaces are equivalent to \(H(\text{curl})\)-Nedelec spaces of the first kind and can be written as \( \begin{cases} [P_{r-1}(\mathbb{R}^2)]^2 \oplus \mathrm{rot}(\bf{x}) H_{r-1}(\mathbb{R}^2), & n = 2, \\ [P_{r-1}(\mathbb{R}^3)]^3 \oplus \bf{x} \times [H_{r-1}(\mathbb{R}^3)]^3, & n = 3. \end{cases} \)

In \(H(\text{div})\) (\(\sim k=n-1\)), trimmed polynomial spaces are equivalent to Raviart-Thomas spaces (\(n=2\)) and \(H(\text{div})\)-Nedelec spaces of the first kind (\(n=3\)), and can be written as \( [P_{r-1}(\mathbb{R}^n)]^n \oplus \bf{x} H_{r-1}(\mathbb{R}^n). \)

In \(L_2\), (\(\sim k=n\)), trimmed polynomial spaces are identical to the standard polynomial spaces of one degree less, \(\mathcal{P}_r^- \sim P_{r-1}\).

## See Also

`PetscSpace`, `PetscSpaceType`, `PetscSpaceCreate()`, `PetscSpaceSetType()`, `PetscDTPTrimmedEvalJet()`

## Level

intermediate

## Location

src/dm/dt/space/impls/ptrimmed/spaceptrimmed.c

Index of all SPACE routines
Table of Contents for all manual pages
Index of all manual pages