Compression techniques for boundary integral equations. Asymptotically optimal complexity estimates.

*(English)*Zbl 1113.65114The paper concerns the wavelet Galerkin discretization of a boundary integral equation with a strongly elliptic and injective operator \(A:H^q(\Gamma)\to H^{-q}(\Gamma)\). The authors provide a complete analysis of the Galerkin schemes with compressed stiffness matrices from the complexity point of view. They propose a new compression strategy with two additional key steps: so-called second compression and a posteriori compression of the stiffness matrices.

The consistency of the compressed scheme with the original operator equation in the corresponding Sobolev norms is established. It is proved that the proposed scheme has the same rate of convergence as the underlying Galerkin method, and moreover, it has optimal complexity. It means that the discretization error accuracy is obtained at a computational expense that is proportional to the size of the arising linear system, uniformly in the size. The derived balance estimates synchronize quadrature and compression accuracy.

The consistency of the compressed scheme with the original operator equation in the corresponding Sobolev norms is established. It is proved that the proposed scheme has the same rate of convergence as the underlying Galerkin method, and moreover, it has optimal complexity. It means that the discretization error accuracy is obtained at a computational expense that is proportional to the size of the arising linear system, uniformly in the size. The derived balance estimates synchronize quadrature and compression accuracy.

Reviewer: Teresa Regińska (Warszawa)

##### MSC:

65N38 | Boundary element methods for boundary value problems involving PDEs |

65Y20 | Complexity and performance of numerical algorithms |

65T60 | Numerical methods for wavelets |

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 |