C 1 Quadratic and C 2 Quartic Macro-Elements on a Modified Morgan-Scott Triangulation

A geometric characterization of Powell-Sabin triangulations allowing the construction of C2 quartic splines

“…In some sense, the characterization obtained here can be seen as a continuation of the work [22]. Indeed, in [22] 𝐶 2 quartic splines on a modified Morgan-Scott refinement is discussed.…”

“…In some sense, the characterization obtained here can be seen as a continuation of the work [22]. Indeed, in [22] 𝐶 2 quartic splines on a modified Morgan-Scott refinement is discussed. The linear functionals involved in the Hermite interpolation problems in [22] and in this paper are the same, only the refinements are different.…”

“…Indeed, in [22] 𝐶 2 quartic splines on a modified Morgan-Scott refinement is discussed. The linear functionals involved in the Hermite interpolation problems in [22] and in this paper are the same, only the refinements are different. Unfortunately, the space developed in [22] is only defined under specific geometrical conditions.…”

“…The linear functionals involved in the Hermite interpolation problems in [22] and in this paper are the same, only the refinements are different. Unfortunately, the space developed in [22] is only defined under specific geometrical conditions. When the three inner points used to define the refinement collapse, this space is not defined, and this is the starting point for the work done in this paper.…”

“…This is because that, the linear system depends on the positions of the triangle split points and the edge split points that determine the modified Morgan-Scott split. The relations between the triangle split points and the edge split points involved in [22] can be viewed as a special case where this linear system has a unique solution.…”