From 271408da9517f0f6fb3fd0485944ea6e9262bad4 Mon Sep 17 00:00:00 2001
From: Francesc Verdugo <f.verdugo.rojano@vu.nl>
Date: Sun, 20 Oct 2024 08:46:18 +0200
Subject: [PATCH] Adding first example

---
 docs/src/examples/problem_types.jl | 48 ++++++++++++++++++++++++++++++
 1 file changed, 48 insertions(+)

diff --git a/docs/src/examples/problem_types.jl b/docs/src/examples/problem_types.jl
index a99644de..92be6172 100644
--- a/docs/src/examples/problem_types.jl
+++ b/docs/src/examples/problem_types.jl
@@ -1 +1,49 @@
 # # Problem types
+#
+# ## Linear, steady-state, single-field
+
+# Solve the following Poisson equation on the unit square,
+#
+# ```math
+# \left\lbrace
+# \begin{aligned}
+# -\Delta u = f  \ &\text{in} \ \Omega,\\
+# u = g \ &\text{on}\ \partial\Omega,\\
+# \end{aligned}
+# \right.
+# ```
+# with $f=1$ and $g=0$.
+#
+# Solve it with a piece-wise linear Lagrange interpolation, and visualize the result.
+
+import GalerkinToolkit as GT
+import ForwardDiff
+import GLMakie
+using LinearAlgebra
+domain = (0,1,0,1)
+cells = (10,10)
+mesh = GT.cartesian_mesh(domain,cells;simplexify=true)
+dirichlet_tag = "dirichlet"
+GT.label_boundary_faces!(mesh;physical_name=dirichlet_tag)
+Ω = GT.interior(mesh)
+Γd = GT.boundary(mesh;physical_names=[dirichlet_tag])
+k = 1
+V = GT.lagrange_space(Ω,k;dirichlet_boundary=Γd)
+uhd = GT.dirichlet_field(Float64,V)
+g = GT.analytical_field(x->0,Ω)
+f = GT.analytical_field(x->1,Ω)
+GT.interpolate_dirichlet!(g,uhd)
+dΩ = GT.measure(Ω,2*k)
+gradient(u) = x->ForwardDiff.gradient(u,x)
+∇(u,x) = GT.call(gradient,u)(x)
+a(u,v) = GT.∫( x->∇(u,x)⋅∇(v,x), dΩ)
+l(v) = GT.∫( x->v(x)*f(x), dΩ)
+x,A,b = GT.linear_problem(uhd,a,l)
+x .= A\b
+uh = GT.solution_field(uhd,x)
+GLMakie.plot(Ω;color=uh,strokecolor=:black)
+
+# !!! warning
+#     TODOs:
+#     - 2D domains should be visualized as 2D plots by default
+#     - Transparent background so that figures look good in dark mode.