Conditions for cubic spline interpolation on triangular elements (Q1059980)

Algebraic conditions which permit one to interpolate twice continuously differentiable piecewise cubic splines over two triangular elements with a common edge are found. Two methods of extending such cubic spline interpolates to arbitrary triangulations are introduced. Section 1 introduces the cubic spline on triangular grids. Let \(\Omega =T_ 1\cup T_ 2\cup...\cup T_ n\) where \(\{T_ i\}\) is a collection of triangles that satisfy a certain condition. Let \(P(x,y)\) be piecewise cubic (not bicubic) polynomials on \(\Omega\). On each \(T_ i\) \(P(x,y)\) has the form \[ P(x,y)=a^ i_ 1+a^ i_ 2x+a^ i_ 3y+a^ i_ 4x^ 2+a^ i_ 5xy+a^ i_ 6y^ 2+a^ i_ 7x^ 3+a^ i_ 8x^ 2y+a^ i_ 9xy^ 2+a^ i_{10}y^ 3. \] \(P(x,y)\) is a cubic spline on \(\Omega\) if \(P(x,y)\in C^ 2(\Omega)\). Section 2 gives the conditions for cubic splines for the case \(n=2\): If \(P(x,y)\) satisfies \(a^ 1_ k=a^ 2_ k\) \((k=1,...,9)\), then \(P(x,y)\in C^ 2(\Omega)\). The interpolation conditions on cubic splines at the sides are given in section 3 and at external nodes \((n=2)\) in section 4. In sections 5 and 6 the conditions of cubic side spline interpolation and cubic node spline interpolation are discussed. The last section provides two examples to show how cubic spline interpolation should proceed on inserted triangles.