Laplacian of trace kernel for internal dofs TNT quadrilateral

According to https://github.com/mscroggs/symfem/blob/main/symfem/elements/tnt.py, we need to multiply by the primary bubble function, and take the laplacian. To this end tabulation which also generates the derivatives is used.

Author: jmv2009

python/demo/demo_tnt-elements.py

@@ -218,10 +218,18 @@ def create_tnt_quad(degree):
         x[2].append(np.zeros([0, 2]))
         M[2].append(np.zeros([0, 1, 0, 1]))
     else:
-        pts, wts = basix.make_quadrature(basix.CellType.quadrilateral, 2 * degree - 1)
-        poly = basix.tabulate_polynomials(
-            basix.PolynomialType.legendre, basix.CellType.quadrilateral, degree - 2, pts
+        pts, wts = basix.make_quadrature(basix.CellType.quadrilateral, 2 * degree)
+        u = pts[:, 0]
+        v = pts[:, 1]
+        pol_set = basix.polynomials.tabulate_polynomial_set(
+            basix.CellType.quadrilateral, basix.PolynomialType.legendre, degree-2, 2, pts
         )
+        # this assumes the conventional [0 to 1][0 to 1] domain of the reference element, 
+        # and takes the Laplacian of (1-u)*u*(1-v)*v*poly, 
+        # cf https://github.com/mscroggs/symfem/blob/main/symfem/elements/tnt.py
+        poly = (pol_set[5]+pol_set[3])*(u-1)*u*(v-1)*v+ \
+                2*(pol_set[2]*(u-1)*u*(2*v-1)+pol_set[1]*(v-1)*v*(2*u-1)+ \
+                   pol_set[0]*((u-1)*u+(v-1)*v))
         face_ndofs = poly.shape[0]
         x[2].append(pts)
         mat = np.zeros((face_ndofs, 1, len(pts), 1))