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Technical Report

Documentation for the HDD method

Alexander Litvinenko


The hierarchical domain decomposition method (HDD method) for solving elliptic differential equations, whose $L^{\infty}$ coefficients may contain a multiscale parameter, was presented in my dissertation work. This technical report describes the main data structures and procedures of the HDD package as well as some examples. The main idea of the HDD method is to build a large scale solution without computing the solution on the small scale. The $\mathcal{H}$-matrix technique yields the efficient $\mathcal{H}$-matrix arithmetic. It is shown that the storage of HDD is $\mathcal{O}(k\sqrt{n_hn_H}\log^2 \sqrt{n_hn_H})$ and the complexity $\mathcal{O (k^2\sqrt{n_hn_H}\log^3 \sqrt{n_hn_H})$, where $k$ is a small rank, $n_h$ and $n_H$ are the numbers of degrees of freedom on fine and coarse grids respectively.

In the case of homogeneous right-hand side HDD has linear storage and complexity $\mathcal{O}(k^2\sqrt{n_hn_H})$.

The method was tested on the so-called skin problem with jumping coefficients and on problems with oscillatory coefficients.

HDD method