On Roundoff Error Growth in Elliptic Problems

Abstract

Large-scale linear systems arise in finite-difference and finite-element discretizations of elliptic problems. With increasing computer performance, ever larger systems are solved using direct methods. How large can such systems be without roundoff compromising accuracy? Here we model roundoff dynamics in standard LU and LDL T decompositions with respect to problem size N . For the one-dimensional (1D) Poisson equation with Dirichlet boundary conditions on an equidistant grid, we show that the relative error in the factorized matrix grows like O(ε √ N ) if roundoffs are modeled as independent, expectation zero random variables. With bias, the growth rate changes to O(ε N ). Subsequent back substitution results in typical error growths of O(ε N √ N ) and O(ε N 2 ), respectively. Error growth is governed by the dynamics of the computational process and by the structure of the boundary conditions rather than by the condition number. Computational results are demonstrated in several examples, including a few fourth-order 1D problems and second-order 2D problems, showing that error accumulation depends strongly on the solution method. Thus, the same LU solver may exhibit different growth rates for the same 2D Poisson problem, depending on whether the five-point or nine-point FDM operator is used.