First-order system least squares for second-order partial differential equations: part I
SIAM Journal on Numerical Analysis - Special issue: the articles in this issue are dedicated to Seymour V. Parter
A convergent adaptive algorithm for Poisson's equation
SIAM Journal on Numerical Analysis
First-Order System Least Squares for Second-Order Partial Differential Equations: Part II
SIAM Journal on Numerical Analysis
A multigrid tutorial (2nd ed.)
A multigrid tutorial (2nd ed.)
An implicit, nonlinear reduced resistive MHD solver
Journal of Computational Physics
Data Oscillation and Convergence of Adaptive FEM
SIAM Journal on Numerical Analysis
Convergence of Adaptive Finite Element Methods
SIAM Review
BoomerAMG: a parallel algebraic multigrid solver and preconditioner
Applied Numerical Mathematics - Developments and trends in iterative methods for large systems of equations—in memoriam Rüdiger Weiss
hypre: A Library of High Performance Preconditioners
ICCS '02 Proceedings of the International Conference on Computational Science-Part III
SIAM Journal on Numerical Analysis
Scalable Parallel Octree Meshing for TeraScale Applications
SC '05 Proceedings of the 2005 ACM/IEEE conference on Supercomputing
SIAM Journal on Scientific Computing
Bottom-Up Construction and 2:1 Balance Refinement of Linear Octrees in Parallel
SIAM Journal on Scientific Computing
Implicit adaptive mesh refinement for 2D reduced resistive magnetohydrodynamics
Journal of Computational Physics
Efficiency-based local adaptive refinement for fosls finite elements
Efficiency-based local adaptive refinement for fosls finite elements
First-Order System Least Squares for Incompressible Resistive Magnetohydrodynamics
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
First-order system least squares and the energetic variational approach for two-phase flow
Journal of Computational Physics
Journal of Computational Physics
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In this paper, we propose new adaptive local refinement (ALR) strategies for first-order system least-squares finite elements in conjunction with algebraic multigrid methods in the context of nested iteration. The goal is to reach a certain error tolerance with the least amount of computational cost and nearly uniform distribution of the error over all elements. To accomplish this, the refinement decision at each refinement level is determined based on optimizing efficiency measures that take into account both error reduction and computational cost. Two efficiency measures are discussed: predicted error reduction and predicted computational cost. These methods are first applied to a two-dimensional (2D) Poisson problem with steep gradients, and the results are compared with the threshold-based methods described in [W. Dörfler, SIAM J. Numer. Anal., 33 (1996), pp. 1106-1124]. Next, these methods are applied to a 2D reduced model of the incompressible, resistive magnetohydrodynamic equations. These equations are used to simulate instabilities in a large aspect-ratio tokamak. We show that, by using the new ALR strategies on this system, we are able to resolve the physics using only 10 percent of the computational cost used to approximate the solutions on a uniformly refined mesh within the same error tolerance.