Nonlinear multigrid based on local spectral coarsening for heterogeneous diffusion problems

Chak S. Lee, François Hamon, Nicola Castelletto, Panayot S. Vassilevski, Joshua A. White

Computer Methods in Applied Mechanics and Engineering


This work develops a nonlinear multigrid method for diffusion problems discretized by cell-centered finite volume methods on general unstructured grids. The multigrid hierarchy is constructed algebraically using aggregation of degrees of freedom and spectral decomposition of reference linear operators associated with the aggregates. For rapid convergence, it is important that the resulting coarse spaces have good approximation properties. In our approach, the approximation quality can be directly improved by including more spectral degrees of freedom in the coarsening process. Further, by exploiting local coarsening and a piecewise-constant approximation when evaluating the nonlinear component, the coarse level problems are assembled and solved without ever re-visiting the fine level, an essential element for multigrid algorithms to achieve optimal scalability. Numerical examples comparing relative performance of the proposed nonlinear multigrid solvers with standard single-level approaches – Picard’s and Newton’s methods – are presented. Results show that the proposed solver consistently outperforms the single-level methods, both in efficiency and robustness.

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Comparison of the overall solution time of single-level Newton vs. several nonlinear multigrid variants for a nonlinear diffusion problem. The parameter alpha controls the nonlinearity of the permeability function. We see the proposed schemes can substantially outperform the classical Newton’s method.