Joshua Nolting Abstract
PhD, May 2008Efficiency-Based Local Adaptive Refinement for FOSLS Finite Elements
Advisor: Tom A. Manteuffel
Adaptive local refinement techniques are heavily utilized to approximate partial differential equations (PDEs) containing local features, including the Poisson equation with steep gradients in the solution, Stokes and Navier-Stokes in domains with re-entrant corners, and magnetohydrodynamics equations with tearing modes. This thesis focuses on a methodology for adaptive local refinement that takes error reduction and computational cost into account to determine which elements to refine. The objective is to attain a prescribed approximation accuracy with the least computational effort. In the context of parallel computer architectures, we also take into account the additional goal of equilibrating the error among elements quickly.
The refinement algorithm is presented with the first-order system least squares (FOSLS) finite element method. The resulting linear systems are solved with algebraic multigrid (AMG). The FOSLS finite element method serves as a viable discretization for adaptive refinement because it yields a, sharp, global a posteriori error estimate and an effective local error indicator. The AMG linear solver couples well with refinement because its convergence is optimal for the FOSLS discretization of many PDEs. If the FOSLS functional is H1-elliptic, then, the AMG convergence factor will be strictly less than one, independent of the finite element mesh-size. At each level, the AMG iterations are initialized with the discrete solution from the previous refinement level, so only a few AMG iterations are typically required there.
