POPEREKENNO-TRIANGULAR METHOD (PTM) FOR SOLVING ELLIPTIC EQUATIONS WITH GRIDS IN AN ARBITRARILY SHAPED FIELD AND ITS MODIFICATION

Abstract

This article considers the issue of efficient solution of elliptic type differential equations generated by the separation method in an arbitrary shaped field. The spectral instability of the Poperekenno-Triangular Method (PTM) on uneven grids is mathematically substantiated. The Modified PTM (MPTM) method with the introduction of a special diagonal operator that smoothes the spectrum is proposed and the necessary conditions for its stable convergence are derived. It is proved that the condition number of the MPTM is independent of the unevenness of the grid steps, and the geometry-independent convergence rate of the iterative process according to the energy norm is estimated. The proposed method allows obtaining stable solutions for arbitrary domains, complex boundary lines, and locally collapsing meshes, and is shown to be an algorithmically efficient approach for large-scale elliptic problems.