Isogeometric Boundary Element Method with Hierarchical Matrices for 3D Helmholtz problems
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In this talk we consider 3D interior and exterior Helmholtz problems, reformulated in terms of a Boundary Integral Equation (BIE). For their numerical solution, we rely on a collocation Boundary Element Method (BEM) formulated in the general framework of Isogeometric Analysis (IgA-BEM), adopting in particular conforming multi-patch discretizations. As it is well known (using BEM as well as IgA-BEM), the matrices of the resulting linear system are fully populated and non-symmetric, a drawback that prevents the application of this strategy to large scale realistic problems. As a possible remedy to reduce the global complexity of the method, we propose a numerical scheme based on the hierarchical matrix (H-matrix) technique. Using a suitable admissibility condition, it starts with hierarchically partitioning the matrix into full- and low-rank blocks. The former are stored and computed in conventional way, meanwhile the latter are approximated by the Adaptive Cross Approximation (ACA) methodology which successfully compresses the dense matrices of the multi-patch IgA-BEM approach. Furthermore, the cost of the matrix-vector product is reduced and this allows us to increase the overall computational efficiency of the Generalized Minimal Residual Method (GMRES), adopted for the solution of the linear system. Several numerical examples are given to demonstrate the accuracy and efficiency of the proposed methodology.
