test(docs): add README doctest snippet

test/docs/readme_example.jl

@@ -0,0 +1,33 @@
+# This file is a part of JuliaFEM.
+# SPDX-License-Identifier: MIT
+
+using SparseArrays
+using Test
+
+"""
+Extract fenced ```julia blocks from README.md (in order).
+"""
+function _readme_julia_blocks(readme_path::AbstractString)
+    readme = read(readme_path, String)
+    blocks = String[]
+    for m in eachmatch(r"(?s)```julia\s*\r?\n(.*?)```", readme)
+        push!(blocks, String(strip(m.captures[1])))
+    end
+    return blocks
+end
+
+@testset "README.md minimal example executes" begin
+    root = joinpath(@__DIR__, "..", "..")
+    readme = joinpath(root, "README.md")
+    @test isfile(readme)
+    blocks = _readme_julia_blocks(readme)
+    @test length(blocks) >= 2
+    # First fence is the Pkg.install snippet; second is "A modern minimal example".
+    M = Module()
+    Base.include_string(M, blocks[2])
+    K = Core.eval(M, :K)
+    f = Core.eval(M, :f)
+    @test size(K) == (375, 375)
+    @test length(f) == 375
+    @test nnz(K) == 19773
+end