An almost optimal method for Dirichlet problems on decomposition subdomains of the hierarchical \(hp\)-version
The discrete Dirichlet problem arising in the domain decomposition method for the interior Dirichlet problems and the preconditioner suggested by \textit{V. G. Korneev} and \textit{S. A. Ivanov} [Mat. Model. 8, No.~9, 63--73 (1996; Zbl 0981.65512)] are considered. The preconditioner is interpretated as a finite-difference operator and is modified on the basis of decomposition of the domain. Exactly, the unit square is decomposed into rectangular subdomains in which the coefficients of the differential equation can be treated as constants without loss of the spectral equivalence. The main ingredients of the fast inversion method for the preconditioner are the fast discrete Fourier transform, the Thomas method for tridiagonal matrices, and the Cholesky elimination method. In the conclusion the total computational complexity of the numerical method is estimated.
scientific article; zbMATH DE number 1758776