Approximate root-free factorization techniques for solving elliptic difference equations in three space variables
In solving a certain selfadjoint elliptic partial differential equation on a cube in \(R^ 3\) by finite difference methods a large sparse linear system of algebraic equations arises with a seven diagonal symmetric positive definite matrix \(\Omega\). For such matrices a direct factorization algorithm is described. With this algorithm we obtain \(\Omega =LDL^ T\), where D is a diagonal matrix, L is a specially structured strictly lower triangular matrix. Choosing two integer parameters \(r_ 1\), \(r_ 2\) an approximate procedure is also described with matrices \(L_ r\), \(D_ r\) and \(\Omega +{\tilde \Pi}=L_ rD_ rL^ T_ r\), where \(L_ r\) contains more zero entries than L. The implicit iterative scheme \(L_ rD_ rL^ T_ r(u_{i+1}-u_ 1)=\alpha (s-\Omega u_ i)\) is further considered. The asymptotic number of iterations is shown to be dependent upon the fill-in parameters \(r_ 1\), \(r_ 2\) and the mesh size h under consideration. Theorem 3.1 of the paper gives an estimate of the number of iterations, required to reduce the \(L_ 2\)-norm of the error by a factor \(\epsilon >0\) and estimate of the total number of required arithmetic operations is given in terms of \(r_ 1\), \(r_ 2\), h.
scientific article; zbMATH DE number 3986576