docs: def _compute_integral_x(self):
          """Computes the integral of x,
  
            \\int_V x,
  
          over all atomic "triangles", i.e., areas cornered by a node, an edge
          midpoint, and a circumcenter.
          """
          # The integral of any linear function over a triangle is the average of
          # the values of the function in each of the three corners, times the
          # area of the triangle.
          right_triangle_vols = self.cell_partitions
  
          node_edges = self.idx_hierarchy
  
          corner = self.node_coords[node_edges]
          edge_midpoints = 0.5 * (corner[0] + corner[1])
          cc = self.cell_circumcenters
  
          average = (corner + edge_midpoints[None] + cc[None, None]) / 3.0
  
          contribs = right_triangle_vols[None, :, :, None] * average
  
          return node_edges, contribs