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Moving least-square reproducing kernel method. II: Fourier analysis. (English) Zbl 0883.65089
Summary: In Part I [ibid. 143, No. 1-2, 113-154 (1997; reviewed above)], the moving least-square reproducing kernel (MLSRK) method is formulated and implemented. Based on its generic construction, an \(m\)-consistency structure is discovered and the convergence theorems are established. In this part, a systematic Fourier analysis is employed to evaluate and further establish the method. The preliminary Fourier analysis reveals that the MLSRK method is stable for sufficiently dense, non-degenerated particle distribution, in the sense that the kernel function family satisfies the Riesz bound. One of the novelties of the current approach is to treat the MLSRK method as a variant of the ‘standard’ finite element method and depart from there to make a connection with the multiresolution approximation. In the spirits of multiresolution analysis, we propose the following MLSRK transformation, \[ {\mathcal F}^{m,k}_{\varrho,h}u= \sum^{np}_{i=1} \langle u,{\mathcal K}_\varrho\rangle_i\check{\mathcal K}^h_\varrho(x- x_i,x)w_i. \] The highlight of this paper is to embrace the MLSRK formulation with the notion of the controlled \(L_p\)-approximation. Based on its characterization, the Strang-Fix condition for example, a systematic procedure is proposed to design new window functions so they can enhance the computational performance of the MLSRK algorithm. The main effort here is to obtain a constant correction function in the interior region of a general domain, i.e. \({\mathcal C}^h_\varrho= 1\). This can create a leap in the approximation order of the MLSRK algorithm significantly, if a highly smooth window function is embedded within the kernel. One consequence of this development is the synchronized convergence phenomenon – a unique convergence mechanism for the MLSRK method, i.e. by properly tuning the dilation parameter, the convergence rate of higher-order error norms will approach the same order convergence rate of the \(L_2\) error norm – they are synchronized.

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65N15 Error bounds for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
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[1] Adams, R.A., Sobolev space, (1975), Academic Press New York, San Francisco
[2] Aubin, Jean-Pierre, Approximation of elliptic boundary-value problems, (1972), John Wiley & Sons · Zbl 0248.65063
[3] Babuǎka, I., Approximation by Hill functions, ()
[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] Chui, C.K., An introduction to wavelets, (1992), Academic Press Boston · Zbl 0925.42016
[6] Carey, G.F.; Oden, J.T., Finite elements: A second course volume II in a five volume series, (1983), Prentice-Hall Englewood Cliffs, NJ
[7] Daubechies, I., Ten lectures on wavelets, society for industrial and applied mathematics, (1992), Philadelphia, Pennsylvania
[8] Davis, P.J., Interpolation & approximation, (1975), Dover Publication New York · Zbl 0111.06003
[9] de Boor, C.; Jia, R.O., Controlled approximation and a characterization of the local approximation order, (), 547-553 · Zbl 0592.41027
[10] Daubechies, I.; Grossmann, A.; Meyer, Y., Painless nonorthogonal expansion, J. math. phys., 27, 1271-1283, (1986) · Zbl 0608.46014
[11] Dahmen, W.; Micchelli, C.A., Translates of multivariate spline, Linear algebra applic., 52/53, 217-234, (1983) · Zbl 0522.41009
[12] Dahmen, W.; Micchelli, C.A., On the approximation order from certain multivariate spline spaces, J. aust. math. soc., series B, 26, 233-246, (1984) · Zbl 0558.41013
[13] Fix, G.; Strang, G., A Fourier analysis of the finite element method in Ritz-Galerkin theory, Studies appl. math., 48, 265-273, (1969) · Zbl 0179.22501
[14] Gomes, S.M.; Cortina, E., Convergence estimates for the wavelet Galerkin method, SIAM J. numer. anal., 33, 149-161, (1996) · Zbl 0845.65048
[15] Jia, R.Q., A counterexample to a result concerning controlled approximation, (), 647-654 · Zbl 0592.41029
[16] Jia, R.Q.; Lei, J., Approximation by multiinteger translates of functions having global support, J. approx. theory, 72, 2-23, (1993) · Zbl 0778.41016
[17] Jia, R.Q.; Lei, J., A new version of the strang-fix conditions, J. approx. theory, 74, 221-225, (1993) · Zbl 0781.41019
[18] Körner, T.W., Fourier analysis, (1988), Cambridge University Press · Zbl 0649.42001
[19] Liu, W.K.; Chen, Y., Wavelet and multiple scale reproducing kernel method, Int. J. numer. methods fluids, 21, 901-933, (1995) · Zbl 0885.76078
[20] Liu, W.K.; Jun, S.; Zhang, S., Reproducing kernel particle methods, Int. J. numer. methods fluids, 20, 1081-1106, (1995) · Zbl 0881.76072
[21] Liu, W-K; Li, S.; Belytschko, T., Moving least square reproducing kernel method, part I: methodology and convergence, Comput. methods appl. mech. engrg., (1995), in press
[22] Meyer, Y., Wavelets and operators, (1992), Cambridge University Press, The French edition was published in 1990 under the name Ondelettes et Operateurs
[23] Papoulis, A., The Fourier integral and its applications, (1962), McGraw-Hill New York · Zbl 0108.11101
[24] Schoenberg, I.J., Contributions to the problem of approximation of equidistant data by analytic functions, part A, Quart. appl. math., 4, 45-99, (1946) · Zbl 0061.28804
[25] Schoenberg, I.J., Contributions to the problem of approximation of equidistant data by analytic functions, part B, Quart. appl. math., 4, 112-141, (1946) · Zbl 0061.28804
[26] Strang, G.; Fix, G., A Fourier analysis of finite element method, () · Zbl 0278.65116
[27] Strang, G., The finite element method and approximation theory, (), 547-584
[28] Strang, G., Approximation in the finite element method, Numer. math, 19, 81-98, (1972) · Zbl 0221.65174
[29] Strichartz, Robert, ()
[30] G. Strang, Private communication, 1996.
[31] Stein, E.M.; Weiss, G., Introduction to Fourier analysis on eucliden spaces, (1971), Princeton University Press Princeton, NJ
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