The ILU method for finite-element discretizations (Q1177224)

As a model problem the author studies the boundary value problem \(\Delta u=g\) in \(\Omega=(0,1)\times(0,1)\), \(u=0\) on \(\partial\Omega\). For finite- difference discretizations on rectangular grids the resulting sparse linear system has a structured pattern; the system can be solved iteratively by incomplete LU decompositions with sparse matrices \(L\) and \(U\) --- called ILU method. This method is transferred to linear systems arising from linear finite element discretizations over problem-oriented triangular grids (self- adaptive refinement). Strategies for acceleration and vectorization of the algorithm are developed (e.g. ringwise numbering of grid points, data structure for the non-zero coefficients, vectorization of uniform parts). It is shown how the numerical stability of the method is influenced by the topology of the grid. For a numerical test an eigenvalue problem of the form \(-\hbox{div}(h \hbox{grad} u)=\lambda u\) in \(\Omega\) (surface of the lake) \(u_ n=0\) on \(\partial\Omega\) (shore of the lake) is treated. It is solved by a multigrid algorithm using ILU iteration as a smoother.