C1 quadratic macroelements and C1 orthogonal multiresolution analyses in 2D

Each triangle of an arbitrary regular triangulation Δ of a polygonal region Ω in R2 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 gD of Δ, C1 quadratic finite elements can be constructed. In this paper, we will give explicit Bézier 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 C1 quadratic spline space under refined grid partition gD can be found. Finally, we discuss the construction of C1 orthogonal scaling functions by using C1 quadratic macroelements.