A quadrature algorithm for wavelet Galerkin methods
- Rathsfeld, Andreas
2010 Mathematics Subject Classification
- 65N38 65T60 65R20 65D30
- wavelet Galerkin methods, first kind integral operator, quadrature algorithm
We consider the wavelet Galerkin method for the solution of boundary integral equations of the first and second kind including integral operators of order r less than zero. This is supposed to be based on an abstract wavelet basis which spans piecewise polynomials of order dT. For example, the bases can be chosen as the basis of tensor product interval wavelets defined over a set of parametrization patches. We define and analyze a quadrature algorithm for the wavelet Galerkin method which utilizes Smolyak quadrature rules of finite order. In particular, we prove that quadrature rules of an order larger than 2dT - r are sufficient to compose a quadrature algorithm for the wavelet Galerkin scheme such that the compressed and quadrature approximated method converges with the maximal order 2dT - r and such that the number of necessary arithmetic operations is less than 𝒪(N log N) with N the number of degrees of freedom. For the estimates, a degree of smoothness greater or equal to 2[2dT - r]+1 is needed.