Starting to draft primal HHO

[amartinhuertas]

WIP ...

[codecov-commenter]

Codecov Report

Attention: Patch coverage is 89.66074% with 64 lines in your changes missing coverage. Please review.

Project coverage is 83.93%. Comparing base (2378958) to head (2f31f13).
Report is 8 commits behind head on main.

❗ Current head 2f31f13 differs from pull request most recent head 40e0463

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Files	Patch %	Lines
src/GridapAPIExtensions.jl	82.19%	13 Missing ⚠️
src/LocalFEOperators.jl	95.01%	13 Missing ⚠️
src/MonomialBasisRefFEs.jl	61.29%	12 Missing ⚠️
src/Skeleton.jl	61.29%	12 Missing ⚠️
src/OrthogonalBasisFESpaces.jl	85.93%	9 Missing ⚠️
src/OrthogonalBasisRefFEs.jl	96.00%	3 Missing ⚠️
src/HybridAffineFEOperators.jl	93.75%	1 Missing ⚠️
src/SkeletonArrays.jl	97.50%	1 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##             main      #20      +/-   ##
==========================================
+ Coverage   81.80%   83.93%   +2.13%     
==========================================
  Files          14       18       +4     
  Lines        2022     2595     +573     
==========================================
+ Hits         1654     2178     +524     
- Misses        368      417      +49     

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[santiagobadia]

# Define HHO local reconstruction operator
m(ur,v) = ∫∇(ur)*∇(v)dΩ
l( (uK,u∂K), v)=∫((∇(v)⋅nK)*u∂K)d∂K+∫(-Δ(v)*uK)dΩ
P=LocalProjector((m,l),UKR,VKR)

# Global formulation
UK∂K = MultiFieldFESpace([UK, U∂K])
a(u,v)=∫(∇(P(v))⋅∇(P(u)))dΩ

function f(v)
   vK, v∂K = UK∂K
   f(v) = ∫(f*vK)dΩ
end

# as HDG

[santiagobadia]

# Stabilisation
# Method 1: Orthogonal basis using change of basis with M^{-1}

refferecᵤ = ReferenceFE(orthogonal,Float64,order+1;space=:P)
\mu_T = h^{-2} # k^4
m(u,v)=∫(∇(v)⋅∇(u))dΩ + \mu_T ∫(v⋅u)dΩ
l( (uK,u∂K), v)=∫((∇(v)⋅nK)*u∂K)d∂K+∫(-Δ(v)*uK)dΩ + \mu_T ∫(v⋅u∂K)dΩ

# Optimisation
# \pi(k,u) that makes 0 dofs higher than k

# Method 2: Define subspaces of orthogonal spaces
# orthogonal basis from 1 to order+1 (excluding 0)
refferecᵤ = ReferenceFE(orthogonal,Float64,1:order+1;space=:P)
refferecᵤ_0 = ReferenceFE(orthogonal,Float64,0;space=:P)
VKR_1 = ...
VKR_0 = ...

VKR = MultiFieldFESpace([UKR_0,UKR_1])

# Define HHO local reconstruction operator
function m(u,v )
  v0, v1 = v
  m(u,v) = ∫∇(u)*∇(v1)dΩ +  ∫(u*v0) dΩ
end

function l(u,v )
  v0, v1 = v
  l( (uK,u∂K), v)=∫((∇(v1)⋅nK)*u∂K)d∂K+∫(-Δ(v1)*uK)dΩ + ∫(uK*v0) dΩ
end

P=LocalProjector((m,l),UKR,VKR)


m(ur,v) = ∫u_mean*vdΩ
l(uK, v)=∫v*uKdΩ
\pi_0=LocalProjector((m,l),UKR,VKR)



# Global formulation
UK∂K = MultiFieldFESpace([UK, U∂K])
a(u,v)=∫(∇(P(v))⋅∇(P(u)))dΩ

function f(v)
   vK, v∂K = UK∂K
   f(v) = ∫(f*vK)dΩ
end
# 


m(u,v)=∫(∇(v)⋅∇(u))dΩ 
l( (uK,u∂K), v)=∫((∇(v)⋅nK)*u∂K)d∂K+∫(-Δ(v)*uK)dΩ 

∫(v⋅u∂K)dΩ

[santiagobadia]

# Stab in Badia et al 2.11
# Define spaces for the projectors

# Cell term
m(ur,v) = ∫u*vdΩ
l( (uK,u∂K), v)=∫(P((uK,u∂K)) - uK)*vdΩ
\PiK =LocalProjector((m,l),UKR,VKR)

# Face term
m(ur,v) = ∫u*vd∂K
l( (uK,u∂K), v)=∫(P((uK,u∂K)) - u∂K)*vd∂K
\Pi∂K =LocalProjector((m,l),UKR,VKR)

# stab term in the global operator
a(u,v)=∫(∇(P(v))⋅∇(P(u)))dΩ + ∫\Pi∂K(u)*\Pi∂K(v)d∂K + ∫\PiK(u)*\PiK(v)dΩ