Adaptive Numerical Treatment of Elliptic Systems on Manifolds

Details Adaptive Numerical Treatment of Elliptic Systems on Manifolds Adaptive Numerical Treatment of Elliptic Systems on Manifolds

2010-01-08

arxiv.org/abs/1001.1367v1

Advances in Computational Mathematics, 15 (2001), pp. 139-191

Mathematics

Numerical Analysis

Version of frequently requested article

Scientific paper

Adaptive multilevel finite element methods are developed and analyzed for certain elliptic systems arising in geometric analysis and general relativity. This class of nonlinear elliptic systems of tensor equations on manifolds is first reviewed, and then adaptive multilevel finite element methods for approximating solutions to this class of problems are considered in some detail. Two a posteriori error indicators are derived, based on local residuals and on global linearized adjoint or dual problems. The design of Manifold Code (MC) is then discussed; MC is an adaptive multilevel finite element software package for 2- and 3-manifolds. It employs a posteriori error estimation, adaptive simplex subdivision, unstructured algebraic multilevel methods, global inexact Newton methods, and numerical continuation methods for the numerical solution of nonlinear covariant elliptic systems on 2- and 3-manifolds. Some of the more interesting features of MC are described in detail, including some new ideas for topology and geometry representation in simplex meshes, and an unusual partition of unity-based method for exploiting parallel computers. A short example is then given which involves the Hamiltonian and momentum constraints in the Einstein equations, a representative nonlinear 4-component covariant elliptic system on a Riemannian 3-manifold which arises in general relativity. A number of operator properties and solvability results recently established are first summarized, making possible two quasi-optimal a priori error estimates for Galerkin approximations which are then derived. These two results complete the theoretical framework for effective use of adaptive multilevel finite element methods. A sample calculation using the MC software is then presented.

Affiliated with

Also associated with

No associations

Rating

Adaptive Numerical Treatment of Elliptic Systems on Manifolds does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Adaptive Numerical Treatment of Elliptic Systems on Manifolds, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Adaptive Numerical Treatment of Elliptic Systems on Manifolds will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-580056