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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 Calderón-Zygmund type are the only assumption on the kernel function.

##### MSC:
 65N38 Boundary element methods for boundary value problems involving PDEs 35S15 Boundary value problems for PDEs with pseudodifferential operators 65T60 Numerical methods for wavelets
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