Fast direct isogeometric boundary element method for 3D potential problems based on HODLR matrix

Abstract A novel fast direct solver based on isogeometric boundary element method (IGABEM) is presented for solving 3D potential problems, which uses the hierarchical off-diagonal low-rank (HODLR) matrix structure arising from the discretization of boundary integral equations. Since the HODLR matrix can be factored into the product form of some diagonal blocks, we can use the Sherman–Morrison–Woodbury formula to solve the inverse of a HODLR matrix efficiently. For large scale problems, an accelerated adaptive cross approximation algorithm is developed to decompose the off-diagonal submatrices. In numerical implementation, bivariate NURBS basis functions are used to describe the geometry. Meanwhile, the same NURBS basis functions are also used to approximate the unknown boundary quantities. The present method is applied to some numerical examples, including an infinite space containing twenty spherical cavities. The numerical results clearly show that the fast direct solver developed in the paper can obtain accurate results with less CPU time.

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