Data-sparse Approximation to the Schur Complement for Laplacian
In this chapter, we discuss a data-sparse approximation to the Schur complement matrix defined on polygonal domains in the case of Laplace equation. First, we introduce the matrix compression techniques in the case of canonical (rectangular) boundaries. It is based on the truncated block-Fourier representation to the Schur complement in 2D (the approach can be easily extended to the case of parallelepiped/cylinder type geometries in 3D). An arbitrary polygonal domain is then decomposed into rectangular and right triangular subdomains by the refined interface. Reducing the FE system of equations to the refined interface and solving the resultant interface equation by multilevel iterations, we arrive at the fast matrix-vector multiplication with the corresponding Schur complement matrix. This yields the linear-logarithmic complexity O(N log3 N) with respect to the number of boundary degrees of freedom N. An extension to the case of refined meshes and exterior domains will be also addressed.
KeywordsNest Selection Sparse Approximation Exterior Problem Interface Equation Nodal Basis Function
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