C¹ Quadratic Macroelements and C¹ Orthogonal Multiresolution Analyses in 2D

Each triangle of an arbitrary regular triangulation Δ of a polygonal region in R ² is subdivided into twelve subtriangles by using three connecting lines joining three arbitrarily chosen points on its edges, three connecting lines from an arbitrarily chosen interior point in the triangle to its three vertices, and three connecting lines joining the points on the edges and the interior point. In this refinement, C ¹ quadratic finite elements can be constructed. In this paper, we will give explicit Bezier coefficients of elements in terms of the parameters that describe function and first partial derivative values at vertices and values of the normal derivatives at vertices of subtriangles that lie on the edges of Δ. Consequently, the basis and approximation properties of C ¹ quadratic spline space under refined grid partition Δ can be found. Finally, we discuss the construction of C ¹ orthogonal scaling functions by using C ¹ quadratic macroelements.