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On C² Quintic Spline Functions Over Triangulations of PowellSabin's Type
 J. COMPUT. APPL. MATH
, 1996
"... Given a triangulation 4 of a polygonal domain, we find a refinement \Delta of 4 by choosing u t in a neighborhood of the center of the inscribed circle of each triangle t 2 4, connecting u t to the vertices of the triangle t, and connecting u t to u t 0 if t 0 2 4 shares an interior edge with t or ..."
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Cited by 9 (3 self)
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Given a triangulation 4 of a polygonal domain, we find a refinement \Delta of 4 by choosing u t in a neighborhood of the center of the inscribed circle of each triangle t 2 4, connecting u t to the vertices of the triangle t, and connecting u t to u t 0 if t 0 2 4 shares an interior edge with t or to the midpoint ve of any boundary edge e of t. The resulting triangulation is a triangulation of PowellSabin's type. We investigate a C² quintic spline space S 2 5 (\Delta) whose elements are C³ only at u t 's. We give a dimension formula for this spline space, show how to construct a locally supported basis, display an interpolation scheme, and prove that this spline space has the full approximation order.
On MultiLevel Bases for Elliptic Boundary Value Problems
 J. Comp. Applied Math
, 1999
"... We study the multilevel method to precondition a linear system arising from discretizing an elliptic partial differential equation of order 2r by using Galerkin's method with spline spaces S 0 1 (4 n ) for r = 1 and S r\Gamma1 3r\Gamma1 (4 n ) or S r\Gamma1 3r\Gamma3 (3 + n ) for r 2, where S ..."
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Cited by 3 (2 self)
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We study the multilevel method to precondition a linear system arising from discretizing an elliptic partial differential equation of order 2r by using Galerkin's method with spline spaces S 0 1 (4 n ) for r = 1 and S r\Gamma1 3r\Gamma1 (4 n ) or S r\Gamma1 3r\Gamma3 (3 + n ) for r 2, where S ae d denotes a spline space of smoothness ae and degree d, 4 n is the nth refinement of a given triangulation 4 0 by using standard uniform regular refinement or nonuniform refinement procedure, and 3 + n is the triangulation obtained from the nth refinement of a given quadrangulation by using uniform or nonuniform refinement procedure. We show we can alway construct a multilevel basis in spline space S 0 1 (4) or S r\Gamma1 3r\Gamma1 (4 n ) or S r\Gamma1 3r\Gamma3 (3 + n ) to precondition the linear system so that its condition number is O((n + 1) 2 ). A detail description of such a construction is given. AMS(MOS) Subject Classifications: 41A15, 41A63, 41A25, 65D10 Keywords and phrases: Bivariate Splines, Bnet, Elliptic Equations, Finite Element Method, Full Approximation Order, Multilevel Basis, Preconditioning.
Lagrange Interpolation by C¹ Cubic Splines on Triangulated Quadrangulations
"... We describe local Lagrange interpolation methods based on C¹ cubic splines on triangulations obtained from arbitrary strictly convex quadrangulations by adding one or two diagonals. Our construction makes use of a fast algorithm for coloring quadrangulations, and the overall algorithm has linear ..."
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Cited by 1 (1 self)
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We describe local Lagrange interpolation methods based on C¹ cubic splines on triangulations obtained from arbitrary strictly convex quadrangulations by adding one or two diagonals. Our construction makes use of a fast algorithm for coloring quadrangulations, and the overall algorithm has linear complexity while providing optimal order approximation of smooth functions.