zbMATH — the first resource for mathematics

On a quadrature algorithm for the piecewise linear wavelet collocation applied to boundary integral equations. (English) Zbl 1074.65141
Summary: We consider a piecewise linear collocation method for the solution of a pseudo-differential equation of order \(r= 0\), \(-1\) over a closed and smooth boundary manifold. The trial space is the space of all continuous and piecewise linear functions defined over a uniform triangular grid and the collocation points are the grid points.
For the wavelet basis in the trial space we choose the three-point hierarchical basis together with a slight modification near the boundary points of the global patches of parametrization. We choose linear combinations of Dirac delta functionals as wavelet basis in the space of test functionals. For the corresponding wavelet algorithm, we show that the parametrization can be approximated by low-order piecewise polynomial interpolation and that the integrals in the stiffness matrix can be computed by quadrature, where the quadrature rules are composite rules of simple low-order quadratures. The whole algorithm for the assembling of the matrix requires no more than \(O(N[\log N]^3)\) arithmetic operations, and the error of the collocation approximation, including in the compression, the approximative parametrization, and the quadratures, is less than \(O(N^{-(2-r)/2})\).
Note that, in contrast to well-known algorithms by T. von Petersdorff and C. Schwab [Numer. Math. 74, No. 4, 479–516 (1996; Zbl 0863.65074)] and of R. Schneider [Multiscale and wavelet matrix compression (1998; Zbl 0899.65063)], only a finite degree of smoothness is required. In contrast to an alllgorithm of S. Ehrich and A. Rathsfeld [ETNA, Electron. Trans. Numer. Anal. 12, 149–192 (2001; Zbl 1002.65144)], no multiplicative splitting of the kernel functions is requied. Beside the usual mapping properties of the integral operator in low order Sobolev spaces, estimates of Calder√≥n-Zygmund type are the only assumption on the kernel function.

65N38 Boundary element methods for boundary value problems involving PDEs
35S15 Boundary value problems for PDEs with pseudodifferential operators
65T60 Numerical methods for wavelets
Full Text: DOI
[1] Beylkin, Communications on Pure and Applied Mathematics 44 pp 141– (1991)
[2] Dahmen, Mathematische Zeitschrift 215 pp 583– (1994)
[3] Dahmen, Advances in Computational Mathematics 1 pp 259– (1993)
[4] Petersdorff, Numerische Mathematik 74 pp 479– (1996)
[5] Fully discrete multiscale Galerkin BEM. In Multiresolution Analysis and PDE, Series: Wavelet Analysis and its Applications, (eds). Academic Press: San Diego, CA, 1997; 287-346.
[6] Boundary element methods with wavelets and mesh refinement. Research Report No. 95-10, SAM, Eidgen?ssische Technische Hochschule Z?rich, 1995.
[7] Multiskalen- und waveletkompression: analysisbasierte Methoden zur effizienten L?sung gro?er vollbesetzter Gleichungssysteme. In Advances in Numerical Mathematics. Teubner: Stuttgart, 1998. · Zbl 0899.65063
[8] Alpert, SIAM Journal on Mathematical Analysis 24 pp 246– (1993)
[9] Harten, SIAM Journal on Numerical Analysis 31 pp 1191– (1994)
[10] Rathsfeld, Journal of Integral Equations and Applications 7 pp 47– (1995) · Zbl 0832.65127
[11] Ehrich, Advances in Computational Mathematics 13 pp 105– (2001)
[12] On a hierarchical three-point basis of piecewise linear functions over smooth surfaces. In Problems and Methods in Mathematical Physics, Series: Operator Theory, Advances and Applications, (eds), vol. 121. Birkh?user Verlag: Basel, 2001; 442-470.
[13] Effiziente L?sung von Gleichungssystemen, die aus der Diskretisierung von schwach singul?ren Integralgleichungen 1. Art herr?hren. Dissertation, Christian-Albrechts-Universit?t, Kiel, 1994.
[14] Stevenson, Numerische Mathematik 80 pp 131– (1998)
[15] Constructing economical Riesz bases for Sobolev spaces. GMD-Bericht, vol. 993. GMD: Sankt Augustin, 1996.
[16] Dahmen, Mathematics of Computation 68 pp 1533– (1999)
[17] Dahmen, SIAM Journal of Mathematical Analysis 31 pp 184– (1999)
[18] Dahmen, Results in Mathematics 34 pp 255– (1998) · Zbl 0931.46006 · doi:10.1007/BF03322055
[19] Canuto, Applied and Computational Harmonic Analysis 6 pp 1– (1999)
[20] Canuto, Applied and Computational Harmonic Analysis 8 pp 123– (1999)
[21] Numerical solution of elliptic problems by the wavelet element method, Preprint, No. 3, Politecnico di Torino, Dipartimento di Matematica, 1989. In Proceedings of the 2nd ENUMATH 97 Conference, et al. (eds). World Scientific: Singapore, 1998; 17-37.
[22] Dahmen, SIAM Journal on Numerical Analysis 37 pp 319– (1999)
[23] Rathsfeld, Journal of Integral Equations and Applications 10 pp 445– (1998) · Zbl 0923.65082
[24] On the stability of piecewise linear wavelet collocation and the solution of the double layer equation over polygonal curves. In Mathematical Fundamentals of the Boundary Element Methods, Advances in Boundry Elements, (ed.). Computational Mechanics Publications: Southampton, 1998; 177-215.
[25] Bourgeois, Mathematical Methods in Applied Sciences 21 pp 1233– (1998)
[26] Lage, SIAM Journal of Science and Computers 20 pp 2195– (1999)
[27] Ehrich, Electronical Transaction on Numerical Analysis 12 pp 149– (2001)
[28] Ten Lectures on Wavelets, CBMS Lecture Notes, vol. 61. SIAM: Philadelphia, 1992. · Zbl 0776.42018 · doi:10.1137/1.9781611970104
[29] Singular Integral Operators. Springer-Verlag: Berlin, Heidelberg, New York, Tokyo, 1986. · doi:10.1007/978-3-642-61631-0
[30] The Numerical Solution of Integral Equations of the Second Kind. Cambridge University Press: Cambridge, New York, Melbourne, 1997. · doi:10.1017/CBO9780511626340
[31] Pr??dorf, Integral Equations and Operator Theory 14 pp 399– (1991)
[32] Pr??dorf, Integral Equations and Operator Theory 15 pp 626– (1992)
[33] Costabel, Numerische Mathematik 62 pp 511– (1992)
[34] Pr??dorf, Journal of Integral Equations and Applications 10 pp 417– (1998)
[35] Rathsfeld, Computer Methods in Applied Mechanical Engineering 157 pp 267– (1998)
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.