Sobolev gradients and boundary conditions for partial differential equations

Neuberger, J. W.

doi:10.1090/conm/204/02629

Cited by 1 publication

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“…We interpret this algorithm in the continuous case as a (projected) Sobolev gradient [23] and show that it preserves the degree. We then give the discretization of the algorithm by P 1 finite element discretization: in numerical interpretation, the algorithm is a (projected) gradient computed with a preconditionned conjugate gradient method.…”

Section: Sobolev Gradient For Harmonic Mapsmentioning

confidence: 99%

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Newton and conjugate gradient for harmonic maps from the disc into the sphere

Pierre

2004

ESAIM: COCV

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Abstract.We compute numerically the minimizers of the Dirichlet energyamong maps u : B 2 → S 2 from the unit disc into the unit sphere that satisfy a boundary condition and a degree condition. We use a Sobolev gradient algorithm for the minimization and we prove that its continuous version preserves the degree. For the discretization of the problem we use continuous P1 finite elements. We propose an original mesh-refining strategy needed to preserve the degree with the discrete version of the algorithm (which is a preconditioned projected gradient). In order to improve the convergence, we generalize to manifolds the classical Newton and conjugate gradient algorithms. We give a proof of the quadratic convergence of the Newton algorithm for manifolds in a general setting.Mathematics Subject Classification. 58E20, 78M10, 65N30, 90C53.

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Section: Sobolev Gradient For Harmonic Mapsmentioning

confidence: 99%

“…By interpreting this algorithm as a Sobolev gradient [23] with step-size 1 on a Riemannian submanifold, we develop a Newton algorithm and conjugate gradient algorithm [20] for Riemannian manifolds. In the appendix, we give a proof of the quadratic convergence of the Newton algorithm for manifolds in a general setting.…”

Section: Introductionmentioning

confidence: 99%

Newton and conjugate gradient for harmonic maps from the disc into the sphere

Pierre

2004

ESAIM: COCV

View full text Add to dashboard Cite

show abstract

Sobolev gradients and boundary conditions for partial differential equations

Cited by 1 publication

References 0 publications

Newton and conjugate gradient for harmonic maps from the disc into the sphere

Newton and conjugate gradient for harmonic maps from the disc into the sphere

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