61 research outputs found
Adaptive Finite Element Methods with Inexact Solvers for the Nonlinear Poisson-Boltzmann Equation
In this article we study adaptive finite element methods (AFEM) with inexact
solvers for a class of semilinear elliptic interface problems. We are
particularly interested in nonlinear problems with discontinuous diffusion
coefficients, such as the nonlinear Poisson-Boltzmann equation and its
regularizations. The algorithm we study consists of the standard
SOLVE-ESTIMATE-MARK-REFINE procedure common to many adaptive finite element
algorithms, but where the SOLVE step involves only a full solve on the coarsest
level, and the remaining levels involve only single Newton updates to the
previous approximate solution. We summarize a recently developed AFEM
convergence theory for inexact solvers, and present a sequence of numerical
experiments that give evidence that the theory does in fact predict the
contraction properties of AFEM with inexact solvers. The various routines used
are all designed to maintain a linear-time computational complexity.Comment: Submitted to DD20 Proceeding
Optimal multilevel methods for H(grad), H(curl), and H(div) systems on graded and unstructured grids
Graph partitioning using matrix values for preconditioning symmetric positive definite systems
Prior to the parallel solution of a large linear system, it is required to
perform a partitioning of its equations/unknowns. Standard partitioning
algorithms are designed using the considerations of the efficiency of the
parallel matrix-vector multiplication, and typically disregard the information
on the coefficients of the matrix. This information, however, may have a
significant impact on the quality of the preconditioning procedure used within
the chosen iterative scheme. In the present paper, we suggest a spectral
partitioning algorithm, which takes into account the information on the matrix
coefficients and constructs partitions with respect to the objective of
enhancing the quality of the nonoverlapping additive Schwarz (block Jacobi)
preconditioning for symmetric positive definite linear systems. For a set of
test problems with large variations in magnitudes of matrix coefficients, our
numerical experiments demonstrate a noticeable improvement in the convergence
of the resulting solution scheme when using the new partitioning approach
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