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Developments in bivarite spline interpolation. (English) Zbl 0960.41006

Authors’ abstract: The aim of this survey is to describe developments in the field of interpolation by bivariate splines. We summarize results on the dimension and the approximation order of bivariate spline spaces, and describe interpolation methods for these spaces. Moreover, numerical examples are given.
Reviewer: E.Deeba (Houston)

MSC:

41A15 Spline approximation
41A05 Interpolation in approximation theory
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References:

[1] M.H. Adam, Bivariate spline-interpolation auf crosscut-partitionen, Dissertation, Mannheim, 1995.; M.H. Adam, Bivariate spline-interpolation auf crosscut-partitionen, Dissertation, Mannheim, 1995.
[2] Alfeld, P., A bivariate \(C^2\) Clough-Tocher scheme, Comput. Aided Geom. Des., 1, 257-267 (1984) · Zbl 0597.65005
[3] Alfeld, P., A trivariate Clough-Tocher scheme for tetrahedral data, Comput. Aided Geom. Des., 1, 169-181 (1984) · Zbl 0566.65003
[4] Alfeld, P., On the dimension of piecewise polynomial functions, (Griffiths, D. F.; Watson, G. A., Numerical Analysis (1986), Longman Scientific & Technical: Longman Scientific & Technical London), 1-23 · Zbl 0655.41010
[5] Alfeld, P., Upper and lower bounds on the dimension of multivariate spline spaces, SIAM J. Numer. Anal., 33, 2, 571-588 (1996) · Zbl 0849.41027
[6] Alfeld, P.; Piper, B.; Schumaker, L. L., Minimally supported bases for spaces of bivariate piecewise polynomials of smoothness \(r\) and degree \(d\)⩾ \(4r+1\), Comput. Aided Geom. Des., 4, 105-123 (1987) · Zbl 0668.41011
[7] Alfeld, P.; Piper, B.; Schumaker, L. L., An explicit basis for \(C^1\) quartic bivariate splines, SIAM J. Numer. Anal., 24, 891-911 (1987) · Zbl 0658.65008
[8] Alfeld, P.; Schumaker, L. L., The dimension of bivariate spline spaces of smoothness \(r\) for degree \(d\)⩾ \(4r+1\), Constr. Approx., 3, 189-197 (1987) · Zbl 0646.41008
[9] Alfeld, P.; Schumaker, L. L., On the dimension of bivariate spline spaces of smoothness \(r\) and degree \(d=3r+1\), Numer. Math., 57, 651-661 (1990) · Zbl 0725.41012
[10] P. Alfeld, L.L. Schumaker, Non-existence of star-supported spline bases, SIAM J. Numer. Anal., to appear, preprint 1999.; P. Alfeld, L.L. Schumaker, Non-existence of star-supported spline bases, SIAM J. Numer. Anal., to appear, preprint 1999. · Zbl 0957.41006
[11] Alfeld, P.; Schumaker, L. L.; Sirvent, M., On dimension and existence of local bases for multivariate spline spaces, J. Approx. Theory, 70, 243-264 (1992) · Zbl 0761.41007
[12] Argyis, J. H.; Fried, I.; Scharpf, D. W., The TUBA family of plate elements for the matrix displacement method, Aeronaut. J. Roy. Aeronaut. Soc., 72, 701-709 (1968)
[13] L. Bamberger, Interpolation in bivariate spline spaces, in: W. Schempp, K. Zeller (Eds.), Multivariate Approximation Theory III, Proceedings of Conference, Oberwolfach, Germany, Birkhäuser, Basel, 1985, pp. 25-34.; L. Bamberger, Interpolation in bivariate spline spaces, in: W. Schempp, K. Zeller (Eds.), Multivariate Approximation Theory III, Proceedings of Conference, Oberwolfach, Germany, Birkhäuser, Basel, 1985, pp. 25-34. · Zbl 0566.41031
[14] Barnhill, R. E.; Farin, G., \(C^1\) quintic interpolation over triangles: two explicit representations, Int. J. Numer. Math. Eng., 17, 1763-1778 (1981) · Zbl 0477.65009
[15] Beatson, R. K.; Ziegler, Z., Monotonicity preserving surface interpolation, SIAM J. Numer. Anal., 22, 2, 401-411 (1985) · Zbl 0579.65011
[16] Bell, K., A refined triangular plate bending element, J. Math. Eng., 1, 101-122 (1969)
[17] Billera, L. J., Homology of smooth splines: generic triangulations and a conjecture of Strang, Trans. Amer. Math. Soc., 310, 2, 325-340 (1988) · Zbl 0718.41017
[18] Billera, L. J.; Haas, R., The dimension and bases of divergence-free splines; a homological approach, Approx. Theory Appl., 7, 1, 91-99 (1991) · Zbl 0758.41013
[19] Boehm, W.; Farin, G.; Kahmann, J., A survey of curve and surface methods in CAGD, Comput. Aided Geom. Des., 1, 1-60 (1984) · Zbl 0604.65005
[20] Bojanov, B. D.; Hakopian, H. A.; Sahakian, A. A., Spline Functions and Multivariate Approximation (1993), Kluwer: Kluwer Dordrecht · Zbl 0772.41011
[21] de Boor, C., A Practical Guide to Splines (1978), Springer: Springer New York · Zbl 0406.41003
[22] de Boor, C., B-form basics, (Farin, G., Geometric Modeling (1987), SIAM: SIAM Philadelphia), 131-148
[23] de Boor, C.; Höllig, K., Approximation order from bivariate \(C^1\) cubics: a counterexample, Proc. AMS, 87, 649-655 (1983) · Zbl 0545.41017
[24] de Boor, C.; Höllig, K., Approximation power of smooth bivariate pp functions, Math. Z., 197, 343-363 (1988) · Zbl 0616.41010
[25] de Boor, C.; Höllig, K.; Riemenschneider, S., Box Splines (1993), Springer: Springer Berlin · Zbl 0814.41012
[26] de Boor, C.; Jia, Q., A sharp upper bound on the approximation order of smooth bivariate pp functions, J. Approx. Theory, 72, 24-33 (1993) · Zbl 0784.41010
[27] Carncier, J. M.; Peña, J. M., A Marsden type identity for periodic trigonometric splines, J. Approx. Theory, 75, 248-255 (1993) · Zbl 0796.41011
[28] Chalmers, B. L.; Leviatan, D.; Prophet, M. P., Optimal interpolating spaces preserving shape, J. Approx. Theory, 98, 354-373 (1999) · Zbl 0952.41021
[29] Chui, C. K., Multivariate Splines, CBMS, Vol. 54 (1988), SIAM: SIAM Philadelphia
[30] Chui, C. K.; He, T. X., On location of sample points in \(C^1\) quadratic bivariate spline interpolation, (Collatz, L.; Meinardus, G.; Nürnberger, G., Numerical Methods of Approximation Theory. Numerical Methods of Approximation Theory, International Series of Numerical Mathematics, Vol. 81 (1987), Birkhäuser: Birkhäuser Basel), 30-43 · Zbl 0676.41006
[31] Chui, C. K.; He, T. X.; Wang, R. H., Interpolation by bivariate linear splines, (Szabados, J.; Tandori, J., Alfred Haar Memorial Conference (1986), North-Holland: North-Holland Amsterdam), 247-255 · Zbl 0622.41003
[32] Chui, C. K.; Hong, D., Construction of local \(C^1\) quartic spline element for optimal-order approximation, Math. Comp., 65, 85-98 (1996) · Zbl 0854.41020
[33] Chui, C. K.; Hong, D., Swapping edges of arbitrary triangulations to achieve the optimal order of approximation, SIAM J. Numer. Anal., 34, 1472-1482 (1997) · Zbl 0889.41014
[34] Chui, C. K.; Hong, D.; Jia, Q., Stability of optimal-order approximation by bivariate splines over arbitrary triangulations, Trans. Amer. Math. Soc., 347, 3301-3318 (1995) · Zbl 0841.41017
[35] Chui, C. K.; Lai, M.-J., On multivariate vertex splines and applications, (Chui, C.; Schumaker, L. L.; Utreras, F., Topics in Multivariate Approximation (1987), Academic Press: Academic Press New York), 19-36
[36] Chui, C. K.; Lai, M.-J., On bivariate super vertex splines, Constr. Approx., 6, 399-419 (1990) · Zbl 0726.41012
[37] Chui, C. K.; Wang, R. H., Multivariate spline spaces, J. Math. Anal. Appl., 94, 197-221 (1983) · Zbl 0526.41027
[38] Chui, C. K.; Wang, R. H., On smooth multivariate spline functions, Math. Comp., 41, 131-142 (1983) · Zbl 0542.41008
[39] Chui, C. K.; Wang, R. H., Spaces of bivariate cubic and quartic splines on type-1 triangulations, J. Math. Anal. Appl., 101, 540-554 (1984) · Zbl 0597.41013
[40] Ciarlet, P. G., Sur l’élément de Clough et Tocher, RAIRO Anal. Numér., 2, 19-27 (1974) · Zbl 0306.65070
[41] Ciarlet, P. G., The Finite Element Method for Elliptic Problems (1977), North-Holland: North-Holland Amsterdam
[42] Ciarlet, P. G., Interpolation error estimates for the reduced Hsieg-Clough-Tocher triangles, Math. Comp., 32, 335-344 (1978) · Zbl 0378.65010
[43] Ciarlet, P. G.; Raviart, P. A., General Lagrange and Hermite interpolation in \(R^n\) with applications to finite element methods, Arch. Rational Mech. Anal., 46, 177-199 (1972) · Zbl 0243.41004
[44] Ciavaldini, J. F.; Nèdélec, J. C., Sur l’ élément de F̂rajis de Veubecke et Sander, RAIRO Anal. Numér, 2, 29-45 (1974) · Zbl 0304.65077
[45] R.W. Clough, J.L. Tocher, Finite element stiffness matries for analysis of plates in bending, Proceedings of Conference on Matrix Methods in Structural Mechanics, Wright Patterson A.F.B., OH, 1965.; R.W. Clough, J.L. Tocher, Finite element stiffness matries for analysis of plates in bending, Proceedings of Conference on Matrix Methods in Structural Mechanics, Wright Patterson A.F.B., OH, 1965.
[46] Costantini, P.; Manni, C., A local shape-preserving interpolation scheme for scattered data, Comput. Aided Geom. Des., 16, 385-405 (1999) · Zbl 0916.68155
[47] W. Dahmen, Bernstein-Bézier representation of polynomial surfaces, Proceedings of ACM SIGGRAPH, Dallas, 1986.; W. Dahmen, Bernstein-Bézier representation of polynomial surfaces, Proceedings of ACM SIGGRAPH, Dallas, 1986. · Zbl 0606.41009
[48] Dahmen, W.; Michelli, C. A., Recent progress in multivariate splines, (Chui, C. K.; Schumaker, L. L.; Ward, J. D., Approximation Theory IV (1983), Academic Press: Academic Press New York), 27-121
[49] Dahmen, W.; Micchelli, C. A., Convexity of multivariate Bernstein polynomials and box spline surfaces, Stud. Sci. Math. Hung., 23, 265-287 (1988) · Zbl 0689.41013
[50] W. Dahmen, C.A. Micchelli, Convexity and Bernstein polynomials on \(k\); W. Dahmen, C.A. Micchelli, Convexity and Bernstein polynomials on \(k\) · Zbl 0805.41004
[51] Dahmen, W.; Gmelig Meyling, R. H.J.; Ursem, J. H.M., Scattered data interpolation by bivariate \(C^1\)-piecewise quadratic functions, Approx. Theory Appl., 6, 6-29 (1990) · Zbl 0724.41003
[52] Davydov, O., On almost interpolation, J. Approx. Theory, 91, 398-418 (1997) · Zbl 0894.41010
[53] Davydov, O., Locally linearly independent basis for \(C^1\) bivariate splines of degree \(q\)⩾ 5, (Daehlen, M.; Lyche, T.; Schumaker, L. L., Mathematical Methods for Curves and Surfaces II (1998), Vanderbilt University Press: Vanderbilt University Press Nashville), 71-77 · Zbl 0904.65014
[54] O. Davydov, G. Nürnberger, Interpolation by \(C^1q\); O. Davydov, G. Nürnberger, Interpolation by \(C^1q\)
[55] Davydov, O.; Nürnberger, G.; Zeilfelder, F., Approximation order of bivariate spline interpolation for arbitrary smoothness, J. Comput. Appl. Math., 90, 117-134 (1998) · Zbl 0932.41002
[56] Davydov, O.; Nürnberger, G.; Zeilfelder, F., Interpolation by cubic splines on triangulations, (Chui, C. K.; Schumaker, L. L., Approximation Theory IX (1998), Vanderbilt University Press: Vanderbilt University Press Nashville), 17-24 · Zbl 0924.41009
[57] Davydov, O.; Nürnberger, G.; Zeilfelder, F., Interpolation by splines on triangulations, (Muller, M. W.; Buhmann, M. D.; Mache, D. H.; Felten, M., New Developments in Approximation Theory. New Developments in Approximation Theory, International Series of Numerical Mathematics, Vol. 132 (1999), Birkhäuser: Birkhäuser Basel), 49-70 · Zbl 0938.41005
[58] O. Davydov, G. Nürnberger, F. Zeilfelder, Cubic spline interpolation on nested polygon triangulations, in: A. Cohen, C. Rabut and L.L. Schumaker (Eds.), Curves and Surfaces, St. Malo, 1999. Vanderbilt University Press, to appear, preprint 1999.; O. Davydov, G. Nürnberger, F. Zeilfelder, Cubic spline interpolation on nested polygon triangulations, in: A. Cohen, C. Rabut and L.L. Schumaker (Eds.), Curves and Surfaces, St. Malo, 1999. Vanderbilt University Press, to appear, preprint 1999.
[59] O. Davydov, G. Nürnberger, F. Zeilfelder, Bivariate spline interpolation with optimal approximation order, Constr. Approx., to appear, preprint 1998.; O. Davydov, G. Nürnberger, F. Zeilfelder, Bivariate spline interpolation with optimal approximation order, Constr. Approx., to appear, preprint 1998.
[60] O. Davydov, L.L. Schumaker, Locally linearly independent bases for bivariate polynomial spline spaces, preprint 1999.; O. Davydov, L.L. Schumaker, Locally linearly independent bases for bivariate polynomial spline spaces, preprint 1999. · Zbl 0971.41005
[61] O. Davydov, L.L. Schumaker, On stable local bases for bivariate polynomial splines, preprint 1999.; O. Davydov, L.L. Schumaker, On stable local bases for bivariate polynomial splines, preprint 1999.
[62] Davydov, O.; Sommer, M.; Strauss, H., On almost interpolation by multivariate splines, (Nürnberger, G.; Schmidt, J. W.; Walz, G., Multivariate Approximation and Splines. Multivariate Approximation and Splines, International Series of Numerical Mathematics, Vol. 125 (1997), Birkhäuser: Birkhäuser Basel), 45-58 · Zbl 0891.41004
[63] Davydov, O.; Sommer, M.; Strauss, H., Locally linearly independent systems and almost interpolation, (Nürnberger, G.; Schmidt, J. W.; Walz, G., Multivariate Approximation and Splines. Multivariate Approximation and Splines, International Series of Numerical Mathematics, Vol. 125 (1997), Birkhäuser: Birkhäuser Basel), 59-72 · Zbl 0894.41002
[64] Davydov, O.; Sommer, M.; Strauss, H., On almost interpolation and locally linearly independent basis, East J. Approx., 5, 67-88 (1999) · Zbl 1084.41501
[65] O. Davydov, M. Sommer, H. Strauss, Interpolation by bivariate linear splines, J. Comput. Appl. Math. 119 (2000) 115-131.; O. Davydov, M. Sommer, H. Strauss, Interpolation by bivariate linear splines, J. Comput. Appl. Math. 119 (2000) 115-131. · Zbl 0965.41004
[66] Diener, D., Instability in the dimension of spaces of bivariate piecewise polynomials of degree \(2r\) and smoothness order \(r\), SIAM J. Numer. Anal., 27, 2, 543-551 (1990) · Zbl 0695.41009
[67] Diereckx, P.; van Leemput, S.; Vermeire, T., Algorithms for surface fitting using Powell-Sabin splines, IMA J. Numer. Anal., 12, 2, 271-299 (1992) · Zbl 0774.65007
[68] Farin, G., A modified Clough-Tocher interpolant, Comput. Aided Geom. Des., 2, 19-27 (1985) · Zbl 0586.65007
[69] Farin, G., Triangular Bernstein-Bézier patches, Comput. Aided Geom. Des., 3, 83-127 (1986)
[70] G. Farin, Geometric Modeling, SIAM, Philadelphia, 1987.; G. Farin, Geometric Modeling, SIAM, Philadelphia, 1987. · Zbl 0636.53002
[71] G. Fraejis de Veubeke, Bending and stretching of plates, Proceedings of Conference on Matrix Methods in Structural Mechanics, Wright Patterson A.F.B., Ohio, 1965.; G. Fraejis de Veubeke, Bending and stretching of plates, Proceedings of Conference on Matrix Methods in Structural Mechanics, Wright Patterson A.F.B., Ohio, 1965.
[72] Gao, J., Interpolation by \(C^1\) quartic bivariate splines, J. Math. Res. Expo., 11, 433-442 (1991) · Zbl 0785.65006
[73] Gao, J., A \(C^2\) finite element and interpolation, Computing, 50, 69-76 (1993) · Zbl 0768.41009
[74] M. Gasca, Th. Sauer, Polynomial interpolation in several variables, Adv. Comput. Math., to appear, preprint 1999.; M. Gasca, Th. Sauer, Polynomial interpolation in several variables, Adv. Comput. Math., to appear, preprint 1999. · Zbl 0943.41001
[75] Gmelig Meyling, R. H.J., Approximation by piecewise cubic \(C^1\)-splines on arbitary triangulations, Numer. Math., 51, 65-85 (1987) · Zbl 0595.41010
[76] Gmelig Meyling, R. H.J.; Pfluger, P. R., On the dimension of the spline space \(S_2^1(Δ)\) in special cases, (Schempp, W.; Zeller, K., Multivariate Approximation Theory III (1985), Birkhäuser: Birkhäuser Basel), 180-190 · Zbl 0567.41015
[77] Gmelig Meyling, R. H.J.; Pfluger, P. R., Smooth interpolation to scattered data by bivariate piecewise polynomials of odd degree, Comput. Aided Geom. Des., 7, 439-458 (1990) · Zbl 0708.65012
[78] Grandine, T. A., An iterative method for computing multivariate \(C^1\) piecewise polynomial interpolants, Comput. Aided Geom. Des., 4, 307-320 (1987) · Zbl 0637.65008
[79] Heindl, G., Interpolation and approximation by piecewise quadratic \(C^1\) functions of two variables, (Schempp, W.; Zeller, K., Multivariate Approximation Theory. Multivariate Approximation Theory, International Series of Numerical Mathematics, Vol. 51 (1979), Birkhäuser: Birkhäuser Basel), 146-161
[80] Hong, D., Spaces of bivariate spline functions over triangulation, Approx. Theory Appl., 7, 56-75 (1991) · Zbl 0756.41017
[81] Hong, D.; Liu, H.; Mohapatra, R., Optimal triangulations and smoothness conditions for bivariate splines, (Chui, C. K.; Schumaker, L. L., Approximation Theory IX (1998), Vanderbilt University Press: Vanderbilt University Press Nashville), 129-136 · Zbl 0928.41008
[82] Ibrahim, A.; Schumaker, L. L., Super spline spaces of smoothness \(r\) and degree \(d\)⩾ \(3r+2\), Constr. Approx., 7, 401-423 (1991) · Zbl 0739.41011
[83] Jeeawock-Zedek, F., Interpolation scheme by \(C^1\) cubic splines on a non-uniform type-2 triangulation of a rectangular domain, C.R. Acad. Sci. Paris Sér. I Math., 314, 413-418 (1992) · Zbl 0754.65014
[84] Jeeawock-Zedek, F., Operator norm and error bounds for interpolating quadratic splines on a non-uniform type-2 triangulation of a rectangular domain, Approx. Theory Appl., 10, 2, 1-16 (1994) · Zbl 0811.41009
[85] Jeeawock-Zebek, F.; Sablonnière, P., Hermite and Lagrange interpolation by quadratic splines on non-uniform criss-cross triangulations, (Laurent, P. J.; etal., Wavelets, Images and Surface Fitting (1994), Addison-Wesley: Addison-Wesley Reading, MA), 445-452 · Zbl 0815.65007
[86] Jensen, T. R.; Toft, B., Graph Coloring Problems (1995), Wiley: Wiley New York · Zbl 0971.05046
[87] Jia, R. Q., Approximation by smooth bivariate splines on a three-direction mesh, (Chui, C. K.; Schumaker, L. L.; Ward, J., Approximation Theory IV (1983), Academic Press: Academic Press New York), 539-545
[88] Jia, R. Q., Approximation order from certain spaces of smooth bivariate splines on a three directional mesh, Trans. Amer. Math. Soc., 295, 199-212 (1986) · Zbl 0637.41017
[89] Jia, R. Q., Local approximation order of box splines, Scentia Sinica, 31, 274-285 (1988) · Zbl 0675.41020
[90] R.Q. Jia, Lower Bounds on the Dimension of Spaces of Bivariate Splines, International Series of Numerical Mathematics, Vol. 94, Birkhäuser, Basel, 1990, pp. 155-165.; R.Q. Jia, Lower Bounds on the Dimension of Spaces of Bivariate Splines, International Series of Numerical Mathematics, Vol. 94, Birkhäuser, Basel, 1990, pp. 155-165.
[91] R.Q. Jia, Lecture Notes on Multivariate Splines, University of Alberta, Edmonton, Canada, 1990.; R.Q. Jia, Lecture Notes on Multivariate Splines, University of Alberta, Edmonton, Canada, 1990.
[92] M. Laghchim-Lahlou, Éléments finis composites de class \(C^kR^2\); M. Laghchim-Lahlou, Éléments finis composites de class \(C^kR^2\) · Zbl 0736.41003
[93] Laghchim-Lahlou, M.; Sablonnière, P., Composite quadrilateral finite elements of class \(C^r\), (Lyche, T.; Schumaker, L. L., Mathematical Methods in CAGD (1989), Academic Press: Academic Press New York), 413-418 · Zbl 0675.41015
[94] Laghchim-Lahlou, M.; Sablonnière, P., \(C^r\) finite elements of HCT, PS and FVS types, (Periaux, J.; Shaw, R. P., Proceedings of the Fifth International Symposium on Numerical Methods in Engineering, Vol. 2 (1989), Springer: Springer Berlin), 163-168 · Zbl 0832.65003
[95] Laghchim-Lahlou, M.; Sablonnière, P., Triangular finite elements of HCT type and class \(C^ρ\), Adv. Comput. Math., 2, 101-122 (1994) · Zbl 0832.65003
[96] Laghchim-Lahlou, M.; Sablonnière, P., Quadrilateral finite elements of FVS type and class \(C^r\), Numer. Math., 30, 229-243 (1995) · Zbl 0824.41012
[97] Laghchim-Lahlou, M.; Sablonnière, P., \(C^r\)-finite elements of Powell-Sabin type on the three directional mesh, Adv. Comput. Math., 6, 191-206 (1996) · Zbl 0867.65002
[98] Lai, M.-J., Approximation order from bivariate \(C^1\)-cubics on a four-directional mesh is full, Comput. Aided Geom. Des., 11, 215-223 (1994) · Zbl 0792.41023
[99] Lai, M.-J., Scattered data interpolation and approximation using bivariate \(C^1\) piecewise cubic polynomials, Comput. Aided Geom. Des., 13, 81-88 (1996) · Zbl 0873.65011
[100] Lai, M.-J., On \(C^2\) quintic spline functions over triangulations of Powell-Sabin’s type, J. Comput. Appl. Math., 73, 135-155 (1996) · Zbl 0866.65007
[101] Lai, M.-J.; Schumaker, L. L., Scattered data interpolation using \(C^2\) supersplines of degree six, SIAM J. Numer. Anal., 34, 905-921 (1997) · Zbl 0872.41004
[102] Lai, M.-J.; Schumaker, L. L., On the approximation power of bivariate splines, Adv. Comput. Math., 9, 251-279 (1998) · Zbl 0924.41010
[103] Lai, M.-J.; Schumaker, L. L., On the approximation power of splines on triangulated quadrangulations, SIAM J. Numer. Anal., 36, 143-159 (1999) · Zbl 0921.41006
[104] M.-J. Lai, L.L. Schumaker, Macro-elements and stable local bases for splines on Clough-Tocher triangulations, preprint 2000.; M.-J. Lai, L.L. Schumaker, Macro-elements and stable local bases for splines on Clough-Tocher triangulations, preprint 2000. · Zbl 0989.65013
[105] M.-J. Lai, L.L. Schumaker, Macro-elements and stable local bases for splines on Powell-Sabin triangulations, preprint 2000.; M.-J. Lai, L.L. Schumaker, Macro-elements and stable local bases for splines on Powell-Sabin triangulations, preprint 2000. · Zbl 1009.41007
[106] Lorente-Pardo, J.; Sablonnière, P.; Serrano-Perez, M. C., Subharmonicity and convexity properties of Bernstein polynomials and Bezier nets on triangles, Comput. Aided Geom. Des., 16, 287-300 (1999) · Zbl 0916.68154
[107] Manni, C., On the dimension of bivariate spline spaces over rectilinear partitions, Approx. Theory Appl., 7, 1, 23-34 (1991) · Zbl 0756.41035
[108] Manni, C., On the dimension of bivariate spline spaces over generalized quasi-cross-cut partitions, J. Approx. Theory, 69, 141-155 (1992) · Zbl 0756.41018
[109] Le Méhauté, A., Construction of surfaces of class \(C^k\) on a domain \(Ω ⊂ \(R^2\) after triangulation, (Schempp, W.; Zeller, K., Multivariate Approximation Theory II (1982), Birkhäuser: Birkhäuser Basel), 223-240
[110] Le Méhauté, A., Unisolvent interpolation in \(R^n\) and the simplicial finite element method, (Chui, C.; Schumaker, L. L.; Utreras, F., Topics in Multivariate Approximation (1987), Academic Press: Academic Press New York), 141-151
[111] C.A. Michelli, Mathematical Aspects of Geometric Modelling, CBMS, Vol. 65, SIAM, Philadelphia, 1995.; C.A. Michelli, Mathematical Aspects of Geometric Modelling, CBMS, Vol. 65, SIAM, Philadelphia, 1995.
[112] Morgan, J.; Scott, R., A nodal basis for \(C^1\) piecewise polynomials of degree \(n\)⩾ 5, Math. Comp., 29, 736-740 (1975) · Zbl 0307.65074
[113] J. Morgan, R. Scott, The dimension of piecewise polynomials, 1977, unpublished manuscript.; J. Morgan, R. Scott, The dimension of piecewise polynomials, 1977, unpublished manuscript.
[114] H. ter Morsche, Bivariate cubic periodic spline interpolation on a three directional mesh, in: L.L. Schumaker et al. (Eds.), Approximation Theory V, Proceedings of Fifth International Symposium, College Station/Texas, 1986, pp. 487-490.; H. ter Morsche, Bivariate cubic periodic spline interpolation on a three directional mesh, in: L.L. Schumaker et al. (Eds.), Approximation Theory V, Proceedings of Fifth International Symposium, College Station/Texas, 1986, pp. 487-490. · Zbl 0639.41006
[115] E. Nadler, Hermite interpolation by \(C^1\); E. Nadler, Hermite interpolation by \(C^1\)
[116] Nürnberger, G., Approximation by Spline Functions (1989), Springer: Springer Berlin · Zbl 0692.41017
[117] G. Nürnberger, Approximation by univariate and bivariate splines, in: D. Bainov, V. Covachev (Eds.), Second International Colloquium on Numerical Analysis, VSP, Utrecht, 1994, pp. 143-155.; G. Nürnberger, Approximation by univariate and bivariate splines, in: D. Bainov, V. Covachev (Eds.), Second International Colloquium on Numerical Analysis, VSP, Utrecht, 1994, pp. 143-155. · Zbl 0845.65002
[118] Nürnberger, G., Approximation order of bivariate spline interpolation, J. Approx. Theory, 87, 117-136 (1996) · Zbl 0864.41010
[119] Nürnberger, G.; Davydov, O.; Walz, G.; Zeilfelder, F., Interpolation by bivariate splines on crosscut partitions, (Nürnberger, G.; Schmidt, J. W.; Walz, G., Multivariate Approximation and Splines. Multivariate Approximation and Splines, International Series of Numerical Mathematics, Vol. 125 (1997), Birkhäuser: Birkhäuser Basel), 189-204 · Zbl 0906.41005
[120] Nürnberger, G.; Rießinger, T., Lagrange and Hermite interpolation by bivariate splines, Numer. Funct. Anal. Optim., 13, 75-96 (1992) · Zbl 0756.41005
[121] Nürnberger, G.; Rießinger, T., Bivariate spline interpolation at grid points, Numer. Math., 71, 91-119 (1995) · Zbl 0831.65008
[122] Nürnberger, G.; Walz, G., Error analysis in interpolation by bivariate \(C^1\)-splines, IMA J. Numer. Anal., 18, 485-508 (1998) · Zbl 0922.41001
[123] Nürnberger, G.; Zeilfelder, F., Spline interpolation on convex quadrangulations, (Chui, C. K.; Schumaker, L. L., Approximation Theory IX (1998), Vanderbilt University Press: Vanderbilt University Press Nashville), 259-266 · Zbl 0933.41009
[124] G. Nürnberger, F. Zeilfelder, Lagrange interpolation by splines on triangulations, in: R.H. Wang (Ed.), Proceedings of the Morningside Institute, Peking, preprint 1999.; G. Nürnberger, F. Zeilfelder, Lagrange interpolation by splines on triangulations, in: R.H. Wang (Ed.), Proceedings of the Morningside Institute, Peking, preprint 1999.
[125] G. Nürnberger, F. Zeilfelder, Interpolation by spline spaces on classes of triangulations, J. Comput. Appl. Math. 119 (2000) 347-376.; G. Nürnberger, F. Zeilfelder, Interpolation by spline spaces on classes of triangulations, J. Comput. Appl. Math. 119 (2000) 347-376. · Zbl 0966.65018
[126] G. Nürnberger, F. Zeilfelder, On bivariate spline spaces, in: W. Hau \(ß\); G. Nürnberger, F. Zeilfelder, On bivariate spline spaces, in: W. Hau \(ß\)
[127] G. Nürnberger, F. Zeilfelder, in preparation.; G. Nürnberger, F. Zeilfelder, in preparation.
[128] G. Nürnberger, L.L. Schumaker, F. Zeilfelder, in preparation.; G. Nürnberger, L.L. Schumaker, F. Zeilfelder, in preparation.
[129] Percell, P., On cubic and quartic Clough-Tocher finite elements, SIAM J. Numer. Anal., 13, 100-103 (1976) · Zbl 0319.65064
[130] Powell, M. J.D., Piecewise quadratic surface fitting for contour plotting, (Evans, D. J., Software for Numerical Analysis (1974), Academic Press: Academic Press New York), 253-277
[131] Powell, M. J.D.; Sabin, M. A., Piecewise quadratic approximation on triangles, ACM Trans. Math. Software, 4, 316-325 (1977) · Zbl 0375.41010
[132] Prenter, P., Splines and Variational Methods (1975), Wiley: Wiley New York · Zbl 0344.65044
[133] Rießinger, T., Interpolation by bivariate quadratic splines on a four-directional mesh, Computing, 49, 129-137 (1992) · Zbl 0762.41003
[134] Ripmeester, D. J., Upper bounds on the dimension of bivariate spline spaces and duality in the plane, (Daehlen, M.; Lyche, T.; Schumaker, L. L., Mathematical Methods for Curves and Surfaces (1995), Vanderbilt University Press: Vanderbilt University Press Nashville), 455-466 · Zbl 0831.41008
[135] de Rose, T.; Goldman, R.; Lounsbery, M., A tutorial introduction to blossoming, (Hagen, H.; Roller, D., Geometric Modelling, Methods and Applications (1991), Springer: Springer Berlin), 267-286
[136] Sablonnière, P., Bernstein-Bézier methods for the construction of bivariate spline approximants, Comput. Aided Geom. Des., 2, 29-36 (1985) · Zbl 0586.65009
[137] Sablonnière, P., Composite finite elements of class \(C^k\), J. Comput. Appl. Math., 12, 541-550 (1985) · Zbl 0587.41004
[138] Sablonnière, P., Composite finite elements of class \(C^2\), (Chui, C. K.; Schumaker, L. L.; Utreras, F. I., Topics in Multivariate Approximation (1987), Academic Press: Academic Press New York), 207-217
[139] Sablonnière, P., Error bounds for Hermite interpolation by quadratic splines on an \(α\)-Triangulation, IMA J. Numer. Anal., 7, 495-508 (1987) · Zbl 0633.41004
[140] Sander, G., Bornes supérieures et inférieures dans l’analyse matricielle des plaques en flexion-torsion, Bull. Soc. Roy. Sci. Liège, 33, 456-494 (1964)
[141] Schmidt, J. W.; Walther, M., Gridded data interpolation with restrictions on the first order derivatives, (Nürnberger, G.; Schmidt, J. W.; Walz, G., Multivariate Approximation and Splines. Multivariate Approximation and Splines, International Series of Numerical Mathematics, Vol. 125 (1997), Birkhäuser: Birkhäuser Basel), 289-305 · Zbl 0889.65006
[142] Schmidt, J. W.; Walther, M., Tensor product splines on refined grids in \(S\)-convex interpolation, (Haußmann, W.; Jetter, K.; Riemer, M.; etal., Multivariate Approximation. Multivariate Approximation, Mathematical Research, Vol. 101 (1997), Akademie Verlag: Akademie Verlag Berlin), 189-202 · Zbl 0897.41009
[143] Schumaker, L. L., On the dimension of piecewise polynomials in two variables, (Schempp, W.; Zeller, K., Multivariate Approximation Theory (1979), Birkhäuser: Birkhäuser Basel), 396-412
[144] Schumaker, L. L., Spline Functions: Basic Theory (1980), Wiley-Interscience: Wiley-Interscience New York · Zbl 0449.41004
[145] Schumaker, L. L., Bounds on the dimension of spaces of multivariate piecewise polynomials, Rocky Mountain J. Math., 14, 251-264 (1984) · Zbl 0601.41034
[146] Schumaker, L. L., Dual bases for spline spaces on a cell, Comput. Aided Geom. Des., 5, 277-284 (1987) · Zbl 0652.41012
[147] Schumaker, L. L., Triangulation methods, (Chui, C. K.; Schumaker, L. L.; Uteras, F. I., Topics in Multivariate Approximation (1987), Academic Press: Academic Press New York), 219-232 · Zbl 0632.65120
[148] Schumaker, L. L., On super splines and finite elements, SIAM J. Numer. Anal., 4, 997-1005 (1989) · Zbl 0679.41008
[149] Seidel, H.-P., An introduction to polar forms, IEEE Comput. Graphics Appl., 13, 1, 38-46 (1993)
[150] Sha, Z., On interpolation by \(S_3^1(Δ_{m,n}^1)\), Approx. Theory Appl., 1, 1-18 (1985)
[151] Sha, Z., On interpolation by \(S_2^1(Δ_{m,n}^2)\), Approx. Theory Appl., 1, 71-82 (1985)
[152] Shi, X. Q., The singularity of Morgan-Scott triangulation, Comput. Aided Geom. Des., 8, 201-206 (1991) · Zbl 0752.41013
[153] Sibson, R.; Thomson, G. D., A seamed quadratic element for contouring, Comput. J., 24, 4, 378-382 (1981)
[154] Sommer, M.; Strauss, H., Interpolation by uni- and multivariate generalized splines, J. Approx. Theory, 83, 423-447 (1995) · Zbl 0837.41010
[155] Sommer, M.; Strauss, H., A condition of Schoenberg-Whitney type for multivariate spline interpolation, Adv. Comput. Math., 5, 381-397 (1996) · Zbl 0856.41002
[156] Strang, G., Piecewise polynomials and the finite element method, Bull. Amer. Math. Soc., 79, 1128-1137 (1973) · Zbl 0285.41009
[157] Strang, G.; Fix, G., An analysis of the Finite Element Method (1973), Prentice-Hall: Prentice-Hall New York · Zbl 0278.65116
[158] Wang, T., A \(C^2\)-quintic spline interpolation scheme on triangulation, Comput. Aided Geom. Des., 9, 379-386 (1992) · Zbl 0770.65005
[159] Whelan, T., A representation of a \(C^2\) interpolant over triangles, Comput. Aided Geom. Des., 3, 53-66 (1986) · Zbl 0626.65006
[160] Whiteley, W., A matrix for splines, (Nevai, P.; Pinkus, A., Progress in Approximation Theory (1991), Academic Press: Academic Press Boston), 821-828
[161] Zedek, F., Interpolation de Lagrange par des splines quadratique sur un quadrilatere de \(R^2\), RAIRO Anal. Numér., 26, 575-593 (1992) · Zbl 0757.41010
[162] Ženišek, A., Interpolation polynomials on the triangle, Numer. Math., 15, 283-296 (1970) · Zbl 0216.38901
[163] Ženišek, A., A general theorem on triangular finite \(C^m\)-elements, RAIRO Anal. Numér., 2, 119-127 (1974) · Zbl 0321.41003
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