Constraint preserving implicit finite element discretization of harmonic map flow into spheres

Bartels, Soeren; Prohl, Andreas

doi:10.1090/s0025-5718-07-02026-1

Cited by 41 publications

(61 citation statements)

References 18 publications

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“…Our numerical experiments for the harmonic map heat flow problem based on Algorithm A do not show significant advantages of the proposed scheme over the algorithms developed in [6,3]. The reason for this is that the fixed point iteration requires in all of our numerical studies that k ≤ h 2 /5, and therefore does not improve existing results.…”

Section: 2mentioning

confidence: 79%

“…In [3,4], a fully practical implicit scheme is given to solve problem (A), which is based on a reformulation of (1.2) using cross products (m = 3),…”

Section: ) Convergent Penalization Strategies Of Problems (A) and (Bmentioning

confidence: 99%

“…We remark that no constraint is required for a different discretization of (1.2)-(1.4) in [3] for the special case where the target is the two-dimensional sphere. For the wave-map equation, conditional convergence towards weak solutions of problem (B) is verified in Theorem 4.1; again, only Brouwer's fixed-point argument to verify existence of iterates requires a mesh-constraint k ≤ Ch max{d/2,1} , while a discrete energy law and the sphere constraint hold unconditionally.…”

Section: Algorithm B Formentioning

confidence: 99%

“…Section 4 correspondingly shows convergence of Algorithm B towards problem (B). Computational experiments to motivate possible blow-up for both problems are reported in Section 5, and are compared with corresponding studies in [3] (harmonic map heat flow), and [7] (wave map equation).…”

Section: Algorithm B Formentioning

confidence: 99%

“…2) Straightforward spatio-temporal discretizations, together with standard finite elements violate the sphere constraint, and lack a discrete energy law; see e.g. [3,15].…”

Section: Introductionmentioning

confidence: 99%

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Convergent discretization of heat and wave map flows to spheres using approximate discrete Lagrange multipliers

Bartels

Lubich²,

Prohl³

2009

Math. Comp.

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Abstract. We propose fully discrete schemes to approximate the harmonic map heat flow and wave maps into spheres. The finite-element based schemes preserve a unit length constraint at the nodes by means of approximate discrete Lagrange multipliers, satisfy a discrete energy law, and iterates are shown to converge to weak solutions of the continuous problem. Comparative computational studies are included to motivate finite-time blow-up behavior in both cases.

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Section: 2mentioning

confidence: 79%

“…In [3,4], a fully practical implicit scheme is given to solve problem (A), which is based on a reformulation of (1.2) using cross products (m = 3),…”

Section: ) Convergent Penalization Strategies Of Problems (A) and (Bmentioning

confidence: 99%

Section: Algorithm B Formentioning

confidence: 99%

Section: Algorithm B Formentioning

confidence: 99%

“…2) Straightforward spatio-temporal discretizations, together with standard finite elements violate the sphere constraint, and lack a discrete energy law; see e.g. [3,15].…”

Section: Introductionmentioning

confidence: 99%

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Convergent discretization of heat and wave map flows to spheres using approximate discrete Lagrange multipliers

Bartels

Lubich²,

Prohl³

2009

Math. Comp.

Self Cite

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Geodesic finite elements on simplicial grids

Sander

2012

Numerical Meth Engineering

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We introduce geodesic finite elements as a conforming way to discretize partial differential equations for functions v W ! M , where is an open subset of R d and M is a Riemannian manifold. These geodesic finite elements naturally generalize standard first-order finite elements for Euclidean spaces. They also generalize the geodesic finite elements proposed for d D 1 in a previous publication of the author. Our formulation is equivariant under isometries of M and, hence, preserves objectivity of continuous problem formulations. We concentrate on partial differential equations that can be formulated as minimization problems. Discretization leads to algebraic minimization problems on product manifolds M n . These can be solved efficiently using a Riemannian trust-region method. We propose a monotone multigrid method to solve the constrained inner problems with linear multigrid speed. As an example, we numerically compute harmonic maps from a domain in R 3 to S 2 .The value space M is, hence, the special Euclidean group SE.3/ D R 3 Ì SO(3). A treatment of onedimensional Cosserat models by geodesic finite elements has been presented in [5]. Finally, PDEs for manifold-valued functions also occur in some image processing applications. For example, the diffusion equation for functions with values in S 2 can be used to denoise color images [6].Various ways have been proposed in the literature to discretize problems for functions with values in a manifold M . The central problem is how to generalize linear interpolation, that is, how to interpolate between a set of values v i 2 M , i D 1, : : : , d C 1. One simple way is to use an embedding of M in a Euclidean space R m [7]. The values are then interpolated in R m and possibly projected back onto M . Although this is simple, and in many cases cheap and even objective, the result depends on the embedding.Alternatively, one can single out a tangent space T p M of M and retract the values v i onto T p M by using the exponential map. The retracted values are then interpolated on T p M and projected back onto M (e.g., [8]). This works only as long as the v i stay away from the cut locus of p. Also, the dependence on a fixed tangent space T p M breaks objectivity.A third approach notes that the interpolation function is needed only at a fixed set of quadrature points. The values there can be treated as additional variables. In the framework of an iterative solver, they are initialized with known values, and only corrections, which live in linear spaces, are ever interpolated. This method was used by Simo and Vu-Quoc [9] to simulate Cosserat rods. However, as was shown later [10], the method introduces spurious dependencies of the solution on the initial iterate and the parameters of the path-following mechanism.In this work, we follow a new approach by using the Riemannian center of mass to generalize linear interpolation. The Riemannian center of mass is well-known in the differential geometry literature (e.g., [11] and the references it contains). Its usefulness for data inte...

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Geometric Finite Elements

Hardering

Sander

2020

Handbook of Variational Methods for Nonlinear Geometric Data

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Constraint preserving implicit finite element discretization of harmonic map flow into spheres

Cited by 41 publications

References 18 publications

Convergent discretization of heat and wave map flows to spheres using approximate discrete Lagrange multipliers

Convergent discretization of heat and wave map flows to spheres using approximate discrete Lagrange multipliers

Geodesic finite elements on simplicial grids

Geometric Finite Elements

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