Minimal Gridap working code used to evaluate equivalence among R J^{-T} and 1/sqrt(det(g)) J R

mwe_grad_perp.jl

using Gridap

# Distorted mesh
function rndm(p::Point)
  r, s = p
  x = r + 0.1*sin(2π*r)*sin(2π*s)
  y = s + 0.1*sin(2π*r)*sin(2π*s)
  Point(x,y)
end

function perp()
  TensorValue{2,2}(0, -1, 1, 0)
end

model   =CartesianDiscreteModel((0,1,0,1),(20,20),map=rndm)
ξₖ      = get_cell_map(model)
Jt      = lazy_map(Broadcasting(∇), ξₖ)
pinvJt  = lazy_map(Operation(Gridap.Fields.pinvJt),Jt)

p=Point(0.5,0.5)
for cell=1:length(Jt)
  Jtp     = evaluate(Jt[cell]    ,p)
  pinvJtp = evaluate(pinvJt[cell],p)
  lhs=perp()⋅pinvJtp                                             # R J^{-T}
  rhs=(1.0/sqrt(det(Jtp⋅transpose(Jtp))))*transpose(Jtp)⋅perp()  # 1/sqrt(det(g)) J R
  @assert norm(lhs-rhs)/norm(rhs) < 1.0e-15
end