Fully Discrete Multiscale Galerkin BEM

Abstract We analyze multiscale Galerkin methods for strongly elliptic boundary integral equations of order zero on closed surfaces in ℝ 3 . Piecewise polynomial, discontinuous multiwavelet bases of any polynomial degree are constructed explicitly. We show that optimal convergence rates in the boundary energy norm and in certain negative norms can be achieved with “compressed” stiffness matrices containing O ( N (log N ) 2 ) nonvanishing entries, where N denotes the number of degrees of freedom on the boundary manifold. We analyze a quadrature scheme giving rise to fully discrete methods. We show that the fully discrete scheme preserves the asymptotic accuracy of the Galerkin scheme with exact integration and without compression. The overall computational complexity of our algorithm is O ( N (log N ) 4 ) kernel evaluations. The implications of the results for the numerical solution of elliptic boundary value problems in or exterior to bounded, three-dimensional domains are discussed.

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