add test of SBP property using new MeshData

test_cutcell_MeshData.jl

@@ -0,0 +1,64 @@
+using NodesAndModes: face_basis
+using Plots
+using LinearAlgebra
+using StaticArrays
+using StartUpDG
+using PathIntersections
+
+N = 3
+quad_rule_face = gauss_lobatto_quad(0, 0, N)
+rd = RefElemData(Quad(), N; quad_rule_face)
+
+cells_per_dimension = 4
+circle = PresetGeometries.Circle(R=0.66, x0=0, y0=0)
+objects = (circle, )
+
+# TODO: remove eventually
+cells_per_dimension_x = cells_per_dimension
+cells_per_dimension_y = cells_per_dimension
+vx = LinRange(-1, 1, cells_per_dimension_x + 1)
+vy = LinRange(-1, 1, cells_per_dimension_y + 1)    
+
+#######################################
+#        test weak SBP property       #
+#######################################
+
+md = StartUpDG.MeshData2(rd, objects, vx, vy; precompute_operators=true)
+mt = md.mesh_type
+
+for (e, elem) in enumerate(mt.physical_frame_elements)
+    xq, yq, wq = md.xq.cut[:,e], md.yq.cut[:,e], md.wJq.cut[:,e]
+
+    Vq, Vxq, Vyq = basis(elem, rd.N, xq, yq)
+    face_ids = mt.cut_face_nodes[e]
+    Vf = vandermonde(elem, rd.N, md.xf.cut[face_ids], md.yf.cut[face_ids]) 
+    Qx, Qy = Vq' * diagm(wq) * Vxq, Vq' * diagm(wq) * Vyq
+
+    (; wJf) = md.mesh_type.cut_cell_data
+    Bx = Diagonal(wJf.cut[face_ids] .* md.nxJ.cut[face_ids])
+    By = Diagonal(wJf.cut[face_ids] .* md.nyJ.cut[face_ids])
+
+    # represent constant vector in basis
+    ee = Vq \ ones(size(Vq, 1))
+    @show norm(ee' * Qx - ee' * Vf' * Bx * Vf)
+    @show norm(ee' * Qy - ee' * Vf' * By * Vf)
+end
+
+# plot pruned cells
+
+# volume quadrature should be exact for degree N(N-1) + 2N-2 polynomials, 
+# or N(N-1) + 2(N-1) = (N+2) * (N-1) degrees
+N_phys_frame_geo = max(2 * N, (N-1) * (N + 2))     
+target_degree = 2 * N
+rd_tri = RefElemData(Tri(), Polynomial(MultidimensionalQuadrature()), N, 
+                    quad_rule_vol=NodesAndModes.quad_nodes_tri(N_phys_frame_geo))
+Np_target = StartUpDG.Np_cut(target_degree)
+
+plot()
+for e in eachindex(cutcells)
+    xq, yq, _ = StartUpDG.subtriangulated_cutcell_quadrature(cutcells[e], rd_tri)
+    scatter!(vec(xq), vec(yq), label="Reference quadrature"); 
+    scatter!(xq_pruned[:,e], yq_pruned[:,e], markersize=8, marker=:circle, 
+             z_order=:back, label="Caratheodory pruning", leg=false)
+end
+display(plot!(leg=false))