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class="is-active"><a href>Darcy (GMG)</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/gridap/GridapSolvers.jl/blob/main/test/Applications/DarcyGMG.jl#" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><pre><code class="language- hljs">module DarcyGMGApplication

using Test
using LinearAlgebra
using FillArrays, BlockArrays

using Gridap
using Gridap.ReferenceFEs, Gridap.Algebra, Gridap.Geometry, Gridap.FESpaces
using Gridap.CellData, Gridap.MultiField, Gridap.Algebra
using PartitionedArrays
using GridapDistributed

using GridapSolvers
using GridapSolvers.LinearSolvers, GridapSolvers.MultilevelTools, GridapSolvers.PatchBasedSmoothers
using GridapSolvers.BlockSolvers: LinearSystemBlock, BiformBlock, BlockTriangularSolver

function get_patch_smoothers(mh,tests,biform,patch_decompositions,qdegree)
patch_spaces = PatchFESpace(tests,patch_decompositions)
nlevs = num_levels(mh)
smoothers = map(view(tests,1:nlevs-1),patch_decompositions,patch_spaces) do tests, PD, Ph
Vh = get_fe_space(tests)
Ω = Triangulation(PD)
dΩ = Measure(Ω,qdegree)
ap = (u,v) -&gt; biform(u,v,dΩ)
patch_smoother = PatchBasedLinearSolver(ap,Ph,Vh)
return RichardsonSmoother(patch_smoother,10,0.2)
end
return smoothers
end

function get_bilinear_form(mh_lev,biform,qdegree)
model = get_model(mh_lev)
Ω = Triangulation(model)
dΩ = Measure(Ω,qdegree)
return (u,v) -&gt; biform(u,v,dΩ)
end

function main(distribute,np,nc,np_per_level)
parts = distribute(LinearIndices((prod(np),)))

Dc = length(nc)
domain = (Dc == 2) ? (0,1,0,1) : (0,1,0,1,0,1)
mh = CartesianModelHierarchy(parts,np_per_level,domain,nc)
model = get_model(mh,1)

order = 2
qdegree = 2*(order+1)
reffe_u = ReferenceFE(raviart_thomas,Float64,order-1)
reffe_p = ReferenceFE(lagrangian,Float64,order-1;space=:P)

u_exact(x) = (Dc==2) ? VectorValue(x[1]+x[2],-x[2]) : VectorValue(x[1]+x[2],-x[2],0.0)
p_exact(x) = 2.0*x[1]-1.0

tests_u = TestFESpace(mh,reffe_u,dirichlet_tags=[&quot;boundary&quot;]);
trials_u = TrialFESpace(tests_u,[u_exact]);
U, V = get_fe_space(trials_u,1), get_fe_space(tests_u,1)
Q = TestFESpace(model,reffe_p;conformity=:L2)

mfs = Gridap.MultiField.BlockMultiFieldStyle()
X = MultiFieldFESpace([U,Q];style=mfs)
Y = MultiFieldFESpace([V,Q];style=mfs)

α = 1.e2
f(x) = u_exact(x) + ∇(p_exact)(x)
graddiv(u,v,dΩ) = ∫(α*divergence(u)⋅divergence(v))dΩ
biform_u(u,v,dΩ) = ∫(v⊙u)dΩ + graddiv(u,v,dΩ)
biform((u,p),(v,q),dΩ) = biform_u(u,v,dΩ) - ∫(divergence(v)*p)dΩ - ∫(divergence(u)*q)dΩ
liform((v,q),dΩ) = ∫(v⋅f)dΩ

Ω = Triangulation(model)
dΩ = Measure(Ω,qdegree)

a(u,v) = biform(u,v,dΩ)
l(v) = liform(v,dΩ)
op = AffineFEOperator(a,l,X,Y)
A, b = get_matrix(op), get_vector(op);

biforms = map(mhl -&gt; get_bilinear_form(mhl,biform_u,qdegree),mh)
patch_decompositions = PatchDecomposition(mh)
smoothers = get_patch_smoothers(
mh,tests_u,biform_u,patch_decompositions,qdegree
)
prolongations = setup_prolongation_operators(
tests_u,qdegree;mode=:residual
)
restrictions = setup_restriction_operators(
tests_u,qdegree;mode=:residual,solver=IS_ConjugateGradientSolver(;reltol=1.e-6)
)

gmg = GMGLinearSolver(
mh,trials_u,tests_u,biforms,
prolongations,restrictions,
pre_smoothers=smoothers,
post_smoothers=smoothers,
coarsest_solver=LUSolver(),
maxiter=3,mode=:preconditioner,verbose=i_am_main(parts)
)

solver_u = gmg
solver_p = CGSolver(JacobiLinearSolver();maxiter=20,atol=1e-14,rtol=1.e-6,verbose=i_am_main(parts))
solver_p.log.depth = 2

bblocks = [LinearSystemBlock() LinearSystemBlock();
LinearSystemBlock() BiformBlock((p,q) -&gt; ∫(-1.0/α*p*q)dΩ,Q,Q)]
coeffs = [1.0 1.0;
0.0 1.0]
P = BlockTriangularSolver(bblocks,[solver_u,solver_p],coeffs,:upper)
solver = FGMRESSolver(20,P;atol=1e-14,rtol=1.e-10,verbose=i_am_main(parts))
ns = numerical_setup(symbolic_setup(solver,A),A)

x = allocate_in_domain(A); fill!(x,0.0)
solve!(x,ns,b)

r = allocate_in_range(A)
mul!(r,A,x)
r .-= b
@test norm(r) &lt; 1.e-5
end

end # module</code></pre><hr/><p><em>This page was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em></p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../NavierStokesGMG/">« Navier-Stokes (GMG)</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.5.0 on <span class="colophon-date" title="Thursday 15 August 2024 14:42">Thursday 15 August 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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