PetscDTAltVPullback#

Compute the pullback of a k-form under a linear transformation of the coordinate space

Synopsis#

#include "petscdt.h" 
PetscErrorCode PetscDTAltVPullback(PetscInt N, PetscInt M, const PetscReal *L, PetscInt k, const PetscReal *w, PetscReal *Lstarw)

Input Parameters#

  • N - the dimension of the origin vector space of the linear transformation, M >= 0

  • M - the dimension of the image vector space of the linear transformation, N >= 0

  • L - a linear transformation, an [M x N] matrix in row-major format

  • k - the signed degree k of the |k|-form w, -(min(M,N)) <= k <= min(M,N). A negative form degree indicates that the pullback should be conjugated by the Hodge star operator (see note).

  • w - a |k|-form in the image space, size [M choose |k|]

Output Parameter#

  • Lstarw - the pullback of w to a |k|-form in the origin space, size [N choose |k|]: (Lstarw)(v_1,…v_k) = w(Lv_1,…,Lv_k).

Note#

Negative form degrees accommodate, e.g., H-div conforming vector fields. An H-div conforming vector field stores its degrees of freedom as (dx, dy, dz), like a 1-form, but its normal trace is integrated on faces, like a 2-form. The correct pullback then is to apply the Hodge star transformation from (M-2)-form to 2-form, pullback as a 2-form, then invert the Hodge star transformation.

See Also#

PetscDTAltV, PetscDTAltVPullbackMatrix(), PetscDTAltVStar()

Level#

intermediate

Location#

src/dm/dt/interface/dtaltv.c

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