On acceleration methods for coupled nonlinear elliptic systems

We compare both numerically and theoretically three techniques for accelerating the convergence of a nonlinear fixed point iteration u→T(u), arising from a coupled elliptic system: Chebyshev acceleration, a second order stationary method, and a nonlinear version of the Generalized Minimal Residual Algorithm (GMRES) which we call NLGMR. All three approaches are implemented in 'Jacobian-free' mode, i.e., only a subroutine which returns T(u) as a function of u is required. We present a set of numerical comparisons for the drift-diffusion semiconductor model. For the mapping T which corresponds to the nonlinear block Gauß-Seidel algorithm for the solution of this nonlinear elliptic system, NLGMR is found to be superior to the second order stationary method and the Chebychev acceleration. We analyze the local convergence of the nonlinear iterations in terms of the spectrum σ[T_u(u^(*))] of the derivative T_u at the solution u^(*). The convergence of the original iteration is governed by the spectral radius ρ{variant}[T_U(u^(*))]. In contrast, the convergence of the two second order accelerations are related to the convex hull of σ[T_u(u^(*))], while the convergence of the GMRES-based approach is related to the local clustering in σ[I-T_u(u^(*))]. The spectrum σ[I-T_u(u^(*))] clusters only at 1 due to the successive inversions of elliptic partial differential equations in T. We explain the observed superiority of GMRES over the second order acceleration by its ability to take advantage of this clustering feature, which is shared by similar coupled nonlinear elliptic systems.