Hdiv BGP formulation

NEWS.md

@@ -6,6 +6,10 @@ and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0
 
 ## [Unreleased]
 
+### Added
+
+- Added drivers for Darcy flow using the BGP method. Since PR[#122](https://github.com/gridap/GridapEmbedded.jl/pull/122).
+
 ### Fixed
 
 - Fix for issue #115, which is a bug in the logic of the `ACTIVE` mask and `GhostSkeleton` construction. Since PR [#123](https://github.com/gridap/GridapEmbedded.jl/pull/123).

test/GridapEmbeddedTests/DarcyBGPflux.jl

@@ -0,0 +1,169 @@
+module DarcyBGPflux
+# Darcy problem: BGP stabilisation with flux boundary conditions (pure pressure trace).
+# A Lagrange multiplier fixes the mean pressure to close the system.
+#
+# Ref: Badia, Boschman, Martín, Nilsson, Ruiz-Baier, Zahedi (2026)
+#      "Divergence-free unfitted finite element discretisations for the Darcy problem"
+#      arXiv:2603.26212
+
+using Test
+using Gridap
+using GridapEmbedded
+using Gridap.Geometry, Gridap.Arrays
+using Gridap.FESpaces, Gridap.CellData
+
+function get_aggregates(cutgeo)
+  geo = get_geometry(cutgeo)
+  strategy = AggregateAllCutCells()
+  aggregates = aggregate(strategy,cutgeo,geo)
+  return Table(filter(agg -> length(agg) > 1, inverse_table(aggregates)))
+end
+
+function projection_operator(ptopo, V, W, Ω, dΩ)
+  Π(u,Ω) = change_domain(u,Ω,DomainStyle(u))
+  mass(u,v) = ∫(u⋅Π(v,Ω))dΩ
+  P = LocalOperator(
+    LocalSolveMap(), ptopo, W, V, mass, mass
+  )
+  return P
+end
+
+function div_projection_operator(ptopo, V, W, Ω, dΩ)
+  Π(u,Ω) = change_domain(u,Ω,DomainStyle(u))
+  lhs(u,v) = ∫(u⋅Π(v,Ω))dΩ
+  rhs(u,v) = ∫((∇⋅u)⋅Π(v,Ω))dΩ
+  P = LocalOperator(
+    LocalSolveMap(), ptopo, W, V, lhs, rhs
+  )
+  return P
+end
+
+function generate_mesh(n,R=0.4,x0=0.835)
+  geo1 = disk(R;x0=Point(0.5,0.5),name="disk")
+  geo2 = plane(x0=Point(x0,0.5),v=Point(1.0,0.0),name="plane")
+  geo = intersect(geo1,geo2,name="domain")
+
+  bgmodel = CartesianDiscreteModel((0,1,0,1),(n,n))
+  cutgeo = cut(bgmodel,geo)
+  return cutgeo
+end
+
+function driver(n,order,u_exact,p_exact,η=1.0,γ=10.0,τ=1.0)
+  cutgeo = generate_mesh(n)
+  bgmodel = get_background_model(cutgeo)
+  qdegree = 2*(order+1)
+
+  Ω_ac = Triangulation(cutgeo,ACTIVE)
+  V = FESpace(Ω_ac,ReferenceFE(raviart_thomas,Float64,order); conformity=:Hdiv)
+  Q = FESpace(Ω_ac,ReferenceFE(lagrangian,Float64,order); conformity=:L2)
+  Λ = ConstantFESpace(Ω_ac)
+  X = MultiFieldFESpace([V,Q,Λ])
+
+  ptopo = PatchTopology(get_grid_topology(bgmodel), get_aggregates(cutgeo))
+  Ωp = PatchTriangulation(bgmodel, ptopo)
+
+  Ω = Triangulation(cutgeo,PHYSICAL)
+  Γu = EmbeddedBoundary(cutgeo)
+
+  dΩ = Measure(Ω, qdegree)
+  dΩp = Measure(Ωp, qdegree)
+  dΓu = Measure(Γu, 2*qdegree)
+
+  Wd = FESpaces.PatchFESpace(bgmodel,Ωp,Float64,order;space=:RT)
+  Πd = projection_operator(ptopo, V, Wd, Ωp, dΩp)
+  W0 = FESpaces.PatchFESpace(bgmodel,Ωp,Float64,order;space=:Q)
+  Π0 = projection_operator(ptopo, Q, W0, Ωp, dΩp)
+  Π0div = div_projection_operator(ptopo, V, W0, Ωp, dΩp)
+
+  f(x) = η*u_exact(x) + ∇(p_exact)(x)
+  g(x) = -(∇⋅u_exact)(x)
+
+  h = γ/n
+  n_Γu = get_normal_vector(Γu)
+  a(u,v) = ∫(η*(u⋅v))dΩ + ∫(h*(u⋅n_Γu)*(v⋅n_Γu))dΓu
+  b(u,q) = ∫(-(∇⋅u)*q)dΩ
+  btilde(u,q) = b(u,q) + ∫((u⋅n_Γu)*q)dΓu
+  c(λ,q) = ∫(λ*q)dΩ
+  l((v,q,μ)) = ∫(v⋅f + g⋅q)dΩ  + ∫(h*(v⋅n_Γu)⋅(u_exact⋅n_Γu))dΓu + c(μ,p_exact)
+
+  sd(u,v) = ∫(τ*(u⋅v))dΩp
+  s0(p,q) = ∫(τ*(p*q))dΩp
+
+  function weakform(x,y)
+    u, p, λ = x
+    v, q, μ = y
+    Πu, Πv = Πd(u), Πd(v)
+    Πp, Πq = Π0(p), Π0(q)
+    divu, divv = ∇⋅u, ∇⋅v
+    Πdivu, Πdivv = Π0div(u), Π0div(v)
+    Xp = FESpaces.PatchFESpace(X,ptopo)
+    data = FESpaces.collect_and_merge_cell_matrix_and_vector(
+      (X, X  , a(u,v) + btilde(v,p) + b(u,q) + c(λ,q) + c(μ,p) + sd(u,v) - s0(divu,q) - s0(divv,p), l(y)),
+      (X, Xp , s0(divu,Πq) + s0(Πdivv,p) - sd(u,Πv), DomainContribution()),
+      (Xp, X , s0(Πdivu,q) + s0(divv,Πp) - sd(Πu,v), DomainContribution()),
+      (Xp, Xp, sd(Πu,Πv) - s0(Πdivu,Πq) - s0(Πdivv,Πp), DomainContribution()),
+    )
+    assem = SparseMatrixAssembler(X,X)
+    A, B = assemble_matrix_and_vector(assem,data)
+    return AffineFEOperator(X,X,A,B)
+  end
+
+  op = weakform(get_trial_fe_basis(X),get_fe_basis(X))
+  uh, ph, λh = solve(op)
+
+  eu = uh-u_exact
+  ep = ph-p_exact
+  l2_err_u = sqrt(sum(∫( eu ⋅ eu )dΩ))
+  l2_err_p = sqrt(sum(∫( ep ⋅ ep )dΩ))
+
+  return cutgeo, uh, ph, l2_err_u, l2_err_p
+end
+
+function convergence(ns,order,u_exact,p_exact)
+  l2_u = zeros(Float64,length(ns))
+  l2_p = zeros(Float64,length(ns))
+  for (i,n) in enumerate(ns)
+    _, _, _, l2_u[i], l2_p[i] = driver(n,order,u_exact,p_exact)
+  end
+  slope_u = log.(l2_u[1:end-1]./l2_u[2:end]) ./ log.(ns[2:end]./ns[1:end-1])
+  slope_p = log.(l2_p[1:end-1]./l2_p[2:end]) ./ log.(ns[2:end]./ns[1:end-1])
+  return l2_u, l2_p, slope_u, slope_p
+end
+
+# Manufactured solution
+u_exact(x) = VectorValue(x[1], -x[2])
+p_exact(x) = x[1] + x[2]
+cutgeo, uh, ph, l2_err_u, l2_err_p = driver(8,1,u_exact,p_exact)
+@test l2_err_u < 1.e-10
+@test l2_err_p < 1.e-10
+
+u_conv(x) = VectorValue(x[1] + sin(π*x[2]), -x[2] + sin(π*x[1]))
+p_conv(x) = sin(π*x[1]) - sin(π*x[2])
+# l2_u, l2_p, su, sp = convergence([8,16,32,64],1,u_conv,p_conv)
+
+writevtk(
+  Ω, "darcy_BGP_flux"; append=false,
+  cellfields=[
+    "uh"=>uh,"ph"=>ph,"u_exact"=>u_exact,"p_exact"=>p_exact,
+    "eu"=>uh-u_exact,"ep"=>ph-p_exact
+  ],
+)
+
+cell_to_agg = flatten_partition(get_aggregates(cutgeo),num_cells(bgmodel))
+writevtk(
+  Triangulation(bgmodel), "aggregates"; append=false,
+  celldata = ["agg"=>cell_to_agg]
+)
+
+function normal_at_centroid(Γ)
+  n = get_normal_vector(Γ)
+  x = CellPoint(map(mean,get_cell_coordinates(Γ)),Γ,PhysicalDomain())
+  return n(x)
+end
+
+Γu = EmbeddedBoundary(cutgeo)
+writevtk(
+  Γu, "boundary_u"; append=false, celldata=["n"=>normal_at_centroid(Γu)]
+)
+
+end # module

test/GridapEmbeddedTests/DarcyBGPmixed.jl

@@ -0,0 +1,174 @@
+module DarcyBGPmixed
+# Darcy problem: BGP stabilisation with mixed boundary conditions (flux + pressure traces
+# imposed on different parts of the unfitted boundary).
+#
+# Ref: Badia, Boschman, Martín, Nilsson, Ruiz-Baier, Zahedi (2026)
+#      "Divergence-free unfitted finite element discretisations for the Darcy problem"
+#      arXiv:2603.26212
+
+using Test
+using Gridap
+using GridapEmbedded
+using Gridap.Geometry, Gridap.Arrays
+using Gridap.FESpaces, Gridap.CellData
+
+function get_aggregates(cutgeo)
+  geo = get_geometry(cutgeo)
+  strategy = AggregateAllCutCells()
+  aggregates = aggregate(strategy,cutgeo,geo)
+  return Table(filter(agg -> length(agg) > 1, inverse_table(aggregates)))
+end
+
+function projection_operator(ptopo, V, W, Ω, dΩ)
+  Π(u,Ω) = change_domain(u,Ω,DomainStyle(u))
+  mass(u,v) = ∫(u⋅Π(v,Ω))dΩ
+  P = LocalOperator(
+    LocalSolveMap(), ptopo, W, V, mass, mass
+  )
+  return P
+end
+
+function div_projection_operator(ptopo, V, W, Ω, dΩ)
+  Π(u,Ω) = change_domain(u,Ω,DomainStyle(u))
+  lhs(u,v) = ∫(u⋅Π(v,Ω))dΩ
+  rhs(u,v) = ∫((∇⋅u)⋅Π(v,Ω))dΩ
+  P = LocalOperator(
+    LocalSolveMap(), ptopo, W, V, lhs, rhs
+  )
+  return P
+end
+
+function generate_mesh(n,R=0.4,x0=0.835)
+  geo1 = disk(R;x0=Point(0.5,0.5),name="disk")
+  geo2 = plane(x0=Point(x0,0.5),v=Point(1.0,0.0),name="plane")
+  geo = intersect(geo1,geo2,name="domain")
+
+  bgmodel = CartesianDiscreteModel((0,1,0,1),(n,n))
+  cutgeo = cut(bgmodel,geo)
+  return cutgeo
+end
+
+function driver(n,order,u_exact,p_exact,η=1.0,γ=10.0,τ=10.0)
+  cutgeo = generate_mesh(n)
+  bgmodel = get_background_model(cutgeo)
+  qdegree = 2*(order+1)
+
+  Ω_ac = Triangulation(cutgeo,ACTIVE)
+  V = FESpace(Ω_ac,ReferenceFE(raviart_thomas,Float64,order); conformity=:Hdiv)
+  Q = FESpace(Ω_ac,ReferenceFE(lagrangian,Float64,order); conformity=:L2)
+  X = MultiFieldFESpace([V,Q])
+
+  ptopo = PatchTopology(get_grid_topology(bgmodel), get_aggregates(cutgeo))
+  Ωp = PatchTriangulation(bgmodel, ptopo)
+
+  Ω = Triangulation(cutgeo,PHYSICAL)
+  Γu = EmbeddedBoundary(cutgeo, "disk","domain")
+  Γp = EmbeddedBoundary(cutgeo, "plane","domain")
+
+  dΩ = Measure(Ω, qdegree)
+  dΩp = Measure(Ωp, qdegree)
+  dΓu = Measure(Γu, 2*qdegree)
+  dΓp = Measure(Γp, 2*qdegree)
+
+  Wd = FESpaces.PatchFESpace(bgmodel,Ωp,Float64,order;space=:RT)
+  Πd = projection_operator(ptopo, V, Wd, Ωp, dΩp)
+  W0 = FESpaces.PatchFESpace(bgmodel,Ωp,Float64,order;space=:Q)
+  Π0 = projection_operator(ptopo, Q, W0, Ωp, dΩp)
+  Π0div = div_projection_operator(ptopo, V, W0, Ωp, dΩp)
+
+  f(x) = η*u_exact(x) + ∇(p_exact)(x)
+  g(x) = -(∇⋅u_exact)(x)
+
+  h = γ/n
+  n_Γu = get_normal_vector(Γu)
+  n_Γp = get_normal_vector(Γp)
+  a(u,v) = ∫(η*(u⋅v))dΩ + ∫(h*(u⋅n_Γu)*(v⋅n_Γu))dΓu
+  b(u,q) = ∫(-(∇⋅u)*q)dΩ
+  btilde(u,q) = b(u,q) + ∫((u⋅n_Γu)*q)dΓu
+  l((v,q)) = ∫(v⋅f + g⋅q)dΩ  + ∫(h*(v⋅n_Γu)⋅(u_exact⋅n_Γu))dΓu - ∫((v⋅n_Γp)*p_exact)dΓp
+
+  sd(u,v) = ∫(τ*(u⋅v))dΩp
+  s0(p,q) = ∫(τ*p*q)dΩp
+
+  function weakform(x,y)
+    u, p = x
+    v, q = y
+    Πu, Πv = Πd(u), Πd(v)
+    Πp, Πq = Π0(p), Π0(q)
+    divu, divv = ∇⋅u, ∇⋅v
+    Πdivu, Πdivv = Π0div(u), Π0div(v)
+    Xp = FESpaces.PatchFESpace(X,ptopo)
+    data = FESpaces.collect_and_merge_cell_matrix_and_vector(
+      (X, X  , a(u,v) + btilde(v,p) + b(u,q) + sd(u,v) - s0(divu,q) - s0(divv,p), l(y)),
+      (X, Xp , s0(divu,Πq) + s0(Πdivv,p) - sd(u,Πv), DomainContribution()),
+      (Xp, X , s0(Πdivu,q) + s0(divv,Πp) - sd(Πu,v), DomainContribution()),
+      (Xp, Xp, sd(Πu,Πv) - s0(Πdivu,Πq) - s0(Πdivv,Πp), DomainContribution()),
+    )
+    assem = SparseMatrixAssembler(X,X)
+    A, B = assemble_matrix_and_vector(assem,data)
+    return AffineFEOperator(X,X,A,B)
+  end
+
+  op = weakform(get_trial_fe_basis(X),get_fe_basis(X))
+  uh, ph = solve(op)
+
+  eu = uh-u_exact
+  ep = ph-p_exact
+  l2_err_u = sqrt(sum(∫( eu ⋅ eu )dΩ))
+  l2_err_p = sqrt(sum(∫( ep ⋅ ep )dΩ))
+
+  return cutgeo, uh, ph, l2_err_u, l2_err_p
+end
+
+function convergence(ns,order,u_exact,p_exact)
+  l2_u = zeros(Float64,length(ns))
+  l2_p = zeros(Float64,length(ns))
+  for (i,n) in enumerate(ns)
+    _, _, _, l2_u[i], l2_p[i] = driver(n,order,u_exact,p_exact)
+  end
+  slope_u = log.(l2_u[1:end-1]./l2_u[2:end]) ./ log.(ns[2:end]./ns[1:end-1])
+  slope_p = log.(l2_p[1:end-1]./l2_p[2:end]) ./ log.(ns[2:end]./ns[1:end-1])
+  return l2_u, l2_p, slope_u, slope_p
+end
+
+# Manufactured solution
+u_exact(x) = VectorValue(x[1], -x[2])
+p_exact(x) = x[1] + x[2]
+cutgeo, uh, ph, l2_err_u, l2_err_p = driver(8,1,u_exact,p_exact)
+@test l2_err_u < 1.e-10
+@test l2_err_p < 1.e-10
+
+u_conv(x) = VectorValue(x[1] + sin(π*x[2]), -x[2] + sin(π*x[1]))
+p_conv(x) = sin(π*x[1]) - sin(π*x[2])
+# l2_u, l2_p, su, sp = convergence([8,16,32,64],1,u_conv,p_conv)
+
+writevtk(
+  Ω, "darcy_BGP_mixed"; append=false, 
+  cellfields=[
+    "uh"=>uh,"ph"=>ph,"u_exact"=>u_exact,"p_exact"=>p_exact,
+    "eu"=>uh-u_exact,"ep"=>ph-p_exact
+  ],
+)
+
+cell_to_agg = flatten_partition(get_aggregates(cutgeo),num_cells(bgmodel))
+writevtk(
+  Triangulation(bgmodel), "aggregates"; append=false,
+  celldata = ["agg"=>cell_to_agg]
+)
+
+function normal_at_centroid(Γ)
+  n = get_normal_vector(Γ)
+  x = CellPoint(map(mean,get_cell_coordinates(Γ)),Γ,PhysicalDomain())
+  return n(x)
+end
+
+Γu = EmbeddedBoundary(cutgeo, "disk","domain")
+Γp = EmbeddedBoundary(cutgeo, "plane","domain")
+writevtk(
+  Γu, "boundary_u"; append=false, celldata=["n"=>normal_at_centroid(Γu)]
+)
+writevtk(
+  Γp, "boundary_p"; append=false, celldata=["n"=>normal_at_centroid(Γp)]
+)
+
+end # module

test/GridapEmbeddedTests/runtests.jl

@@ -20,4 +20,9 @@ using Test
 
 @time @testset "TraceFEM" begin include("TraceFEMTests.jl") end
 
+@time @testset "DarcyBGP" begin 
+  include("DarcyBGPflux.jl")
+  include("DarcyBGPmixed.jl")
+end
+
 end