Moving least-square reproducing kernel methods. I: Methodology and convergence. (English) Zbl 0883.65088

Summary: This paper formulates the moving least-square interpolation scheme in a framework of the so-called moving least-square reproducing kernel (MLSRK) representation. In this study, the procedure of constructing moving least-square interpolation function is facilitated by using the notion of reproducing kernel formulation, which, as a generalization of the early discrete approach, establishes a continuous basis for a partition of unity. This new formulation possesses the quality of simplicity, and it is easy to implement. Moreover, the reproducing kernel formula proposed is not only able to reproduce any \(m\)th order polynomial exactly on an irregular particle distribution, but also serves as a projection operator that can approximate any smooth function globally with an optimal accuracy.
In this contribution, a generic \(m\)-consistency relation has been found, which is the essential property of the MLSRK approximation. An interpolation error estimate is given to assess the convergence rate of the approximation. It is shown that for a sufficiently smooth function the interpolant expansion in terms of sampled values will converge to the original function in the Sobolev norms. As a meshless method, the convergence rate is measured by a new control variable – dilation parameter \(\varrho\) of the window function, instead of the mesh size \(h\) as usually done in the finite element analysis. To illustrate the procedure, convergence has been shown for the numerical solution of the second-order elliptic differential equations in a Galerkin procedure invoked with this interpolant. In the numerical example, a two-point boundary problem is solved by using the method, and an optimal convergence rate is observed with respect to various norms.


65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs


Zbl 0883.65089
Full Text: DOI


[1] Gingold, R. A.; Monaghan, J. J., Kernel estimates as a basis for general particle methods in hydrodynamics, J. Comput. Phys., 46, 429-453 (1982) · Zbl 0487.76010
[2] Monaghan, J. J., Why particle methods work, SIAM J. Scient. Stat. Comput., 3, 422-433 (1982) · Zbl 0498.76010
[3] Nayroles, B.; Touzot, G.; Villon, P., Generalizing the finite element method: Diffuse approximation and diffuse elements, Comput. Mech., 10, 307-318 (1992) · Zbl 0764.65068
[4] Belytschko, T.; Lu, Y. Y.; Gu, L., Element free Galerkin methods, Int. J. Numer. Methods Engrg., 37, 229-256 (1994) · Zbl 0796.73077
[5] Belytschko, T.; Gu, L.; Lu, Y. Y., Fracture and crack growth by element free Galerkin methods, Modelling Simul. Mater. Sci. Engrg., 2, 519-534 (1994)
[6] Liu, W. K.; Jun, S.; Zhang, Y., Reproducing kernel particle methods, Int. J. Numer. Methods Fluids, 20, 1081-1106 (1995) · Zbl 0881.76072
[7] Liu, W. K.; Chen, Y., Wavelet and multiple scale reproducing kernel methods, Int. J. Numer. Methods Fluids, 21, 901-933 (1995) · Zbl 0885.76078
[8] Liu, W. K.; Jun, S.; Li, S.; Adee, J.; Belytschko, T., Reproducing kernel particle methods for structural dynamics, Int. J. Numer. Methods Engrg., 38, 1655-1679 (1995) · Zbl 0840.73078
[9] Liu, W. K., An introduction to wavelet reproducing kernel particle methods, USACM Bulletin, 8, 1, 3-16 (1995)
[10] Lancaster, P.; Salkauskas, K., Surfaces generated by moving least square methods, Math. Comput., 37, 155, 141-158 (1980) · Zbl 0469.41005
[11] Li, S.; Liu, W. K., Moving least square reproducing kernel method. Part II: Fourier analysis, Comput. Methods Appl. Mech. Engrg., 139, 159-194 (1996) · Zbl 0883.65089
[12] Wells, R. O.; Zhou, X., Wavelet interpolation and approximate solutions of elliptic partial differential equations (1993), preprint
[13] Duarte, C. A.; Oden, J. T., Hp Clouds—a meshless method to solve boundary-value problems (1995), preprint
[14] Rudin, W., Principles of Mathematical Analysis (1976), McGraw-Hill, Inc · Zbl 0148.02903
[15] Mitrinovic̀, D. S., Analytic Inequalities (1970), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York · Zbl 0199.38101
[16] Davis, P. J., Interpolation and Approximation (1975), Dover Publications, Inc: Dover Publications, Inc New York · Zbl 0111.06003
[17] Chui, C. K., An Introduction to Wavelet (1992), Academic Press: Academic Press New York
[18] Babuška, I.; Melenk, J. M., The partition of unity finite element method, (Technical Note BN-1185 (1995), Institute for Physical Science and Technology, University of Maryland) · Zbl 0949.65117
[19] de Boor, C.; Fix, G. J., Spline approximation by quasi-interpolants, J. Approx. Theory, 8, 19-45 (1975) · Zbl 0279.41008
[20] Horn, R. A.; Johnson, C. R., Matrix Analysis (1985), Cambridge University Press · Zbl 0576.15001
[21] Golub, G. H.; Van Loan, C. F., Matrix Computations (1989), The Johns Hopkins University Press: The Johns Hopkins University Press Baltimore, MD · Zbl 0733.65016
[22] Ciarlet, P. G., The Finite Element Method For Elliptic Problems (1978), North-Holland Publishing Company: North-Holland Publishing Company Amsterdam, New York · Zbl 0445.73043
[23] Babuška, I., The finite element method with Lagrangian multipliers, Numer. Math., 20, 179-192 (1973) · Zbl 0258.65108
[24] Hughes, T. J.R., The Finite Element Method: Linear Static and Dynamic Finite Element Analysis (1987), Prentice Hall: Prentice Hall Englewood Cliffs, NJ · Zbl 0634.73056
[25] Rachford, H. H.; Wheeler, M. F., An \(H^{−1}\)-Galerkin procedure for the two-point boundary value, (de Boor, C., Mathematical Aspects of Finite Elements in Partial Differential Equations (1974), Academic Press: Academic Press New York) · Zbl 0347.65037
[26] Babuška, I.; Oden, J. T.; Lee, J. K., Mixed-hybrid finite element approximations of second-order elliptic boundary-value problems, Comput. Methods Appl. Mech. Engrg., 11, 175-206 (1977) · Zbl 0382.65056
[27] Daubechies, I., Wavelets, CBMS-NSF Series in Applied Mathematics (1992), SIAM Publication: SIAM Publication Philadelphia
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.