Lanczos and Arnoldi methods for the solution of convection-diffusion equations

Abstract

Two procedures for computing the finite element solution to a convection-diffusion equation, based on the Lanczos and Arnoldi methods, are presented. The Lanczos vectors are generated from the symmetric part of the coefficient matrices, while the Arnoldi vectors are obtained from the unsymmetric matrix. Following a Rayleigh-Ritz procedure, these vectors are then used to reduce the governing system of differential equations to a small system. The Lanczos process produces symmetric tridiagonal and skew-symmetric matrix coefficients for the reduced system. The Arnoldi method results in an upper Hessenberg matrix coefficient for its reduced problem. The finite element solution is then constructed as a linear combination of the Lanczos or Arnoldi vectors. The solution vector for the reduced system holds the components of the approximating solution along the Lanczos or Arnoldi vectors. Each method is applied to a number of different numerical test problems. We conclude that the Lanczos method is preferred for diffusion dominated problems, while the Arnoldi is the method of choice for convection dominated problems.