Convergence of the triharmonic spline method Cooper Cunliffe Department of Mathematics University of North Carolina Asheville Asheville NC 28804 Convergence of the tri harmonic spline method p 1 13 The Tri Harmonic Spline Given the data set xi yi fi i 1 V with fi f xi yi we consider the spline space Sdr s C r s t Pd t Convergence of the tri harmonic spline method p 2 13 The Tri Harmonic Spline Given the data set xi yi fi i 1 V with fi f xi yi we consider the spline space Sdr s C r s t Pd t d degree of the spline space r smoothness number of times differentiable a triangulation of the data sites xi yi i 1 V the union of all triangles in Pd the space of all polynomials of degree d Convergence of the tri harmonic spline method p 2 13 The Tri Harmonic Spline we are looking for the spline Sf Sdr such that Sf xi yi fi i 1 V and H Sf min H s s Sdr where H s XZ T T Dx3 s 2 3 Dx2 Dy s 2 3 Dx Dy2 s 2 3 2 Dy s dxdy Convergence of the tri harmonic spline method p 3 13 Purpose of Proving Convergence We want to show that Sf will converge to the data function f as the number of data sites increases Convergence of the tri harmonic spline method p 4 13 The Convergence Theorem Let Sf be the spline interpolating f at the vertices of Suppose that f C 3 Then there exists a constant C dependent on d and as well as the Lipschitz constant associated with the boundary if is not convex such that kf Sf kL2 C 3 f 3 Convergence of the tri harmonic spline method p 5 13 Lemmas Lemma 1 Convergence of the tri harmonic spline method p 6 13 Lemmas Lemma 1 Given a triangle T in and domain T then for every f Wqm 1 T with 0 m d and 1 q kDx Dy f Qf kq T K T m 1 f m 1 q T for all 0 m Convergence of the tri harmonic spline method p 6 13 Lemmas Lemma 1 Given a triangle T in and domain T then for every f Wqm 1 T with 0 m d and 1 q kDx Dy f Qf kq T K T m 1 f m 1 q T for all 0 m Lemma 2 Convergence of the tri harmonic spline method p 6 13 Lemmas Lemma 1 Given a triangle T in and domain T then for every f Wqm 1 T with 0 m d and 1 q kDx Dy f Qf kq T K T m 1 f m 1 q T for all 0 m Lemma 2 Suppose that g is continuously three times differentiable over a triangle T Suppose that g is zero at six vertices in Star T which do not lie on a conic section Then kgkL T C1 T 3 g 3 T for a positive constant C1 independent of g and T Convergence of the tri harmonic spline method p 6 13 Lemmas Lemma 3 Convergence of the tri harmonic spline method p 7 13 Lemmas Lemma 3 Let T be a triangle and let AT be its area Then for all p Pd and all 1 q 1 q p T KAT p q T If we pick q 2 K C2 and p Sf we get C2 Sf 3 T Sf 3 2 T AT where Sf 3 2 T sZ T Dx3 Sf 2 3 Dx2 Dy Sf 2 3 Dx Dy2 Sf 2 Dy3 Sf 2 dxdy Convergence of the tri harmonic spline method p 7 13 Proof of Convergence Since by definition Sf f 0 at the vertices of T we can apply Lemma 2 and get Sf f C1 T 3 Sf f 3 T Also note that H Sf X T Sf 23 2 T Convergence of the tri harmonic spline method p 8 13 Proof of Convergence Thus we have Z Sf f 2 dxdy XZ T T Sf f 2 dxdy Convergence of the tri harmonic spline method p 9 13 Proof of Convergence Thus we have Z Sf f 2 dxdy C1 X T XZ T T Sf f 2 dxdy T 6 AT Sf f 23 T Convergence of the tri harmonic spline method p 9 13 Proof of Convergence Thus we have Z Sf f 2 dxdy C1 6 C1 X T X T XZ T T Sf f 2 dxdy T 6 AT Sf f 23 T AT f 3 T Sf 3 T 2 Convergence of the tri harmonic spline method p 9 13 Proof of Convergence Thus we have Z Sf f 2 dxdy C1 6 C1 6 C1 X T AT X T X T XZ T T Sf f 2 dxdy T 6 AT Sf f 23 T AT f 3 T Sf 3 T 2 f 23 T 2 f 3 T Sf 3 T 2 Sf 3 T Convergence of the tri harmonic spline method p 9 13 Proof of Convergence 6 C1 X T AT f 23 T 2 f 3 T Sf 3 T Sf 23 T Convergence of the tri harmonic spline method p 10 13 Proof of Convergence 6 C1 6 C1 X AT f 23 T 2 f 3 T Sf 3 T Sf 23 T X AT f 23 T f 23 T Sf 23 T Sf 23 T T T Convergence of the tri harmonic spline method p 10 13 Proof of Convergence 6 C1 6 C1 X AT f 23 T 2 f 3 T Sf 3 T Sf 23 T X AT f 23 T f 23 T Sf 23 T Sf 23 T T T 6 C1 X T AT 2 f 23 T 2 2 Sf 3 T Convergence of the tri harmonic spline method p 10 13 Proof of Convergence 6 C1 6 C1 X AT f 23 T 2 f 3 T Sf 3 T Sf 23 T X AT f 23 T f 23 T Sf 23 T Sf 23 T T T 6 C1 X T AT 2 f 23 T 2 2 Sf 3 T C 2 2C1 6 AT f 23 T Sf 3 2 T 2 AT T X Convergence of the tri harmonic spline method p 10 13 Proof of Convergence C 2 2C1 6 AT f 23 T Sf 3 2 T 2 AT T X Convergence of the tri harmonic spline method p 11 13 Proof of Convergence C 2 2C1 6 AT f 23 T Sf 3 2 T 2 AT T X 2C1 6 X T 2 C AT f 23 T 2 Sf 23 2 T AT Convergence of the tri harmonic spline method p 11 13 Proof of Convergence C 2 2C1 6 AT f 23 T Sf 3 2 T 2 AT T X 2C1 6 6 X T 0 2 C AT f 23 T 2 Sf …