Geodesic Finite Elements on Simplicial Grids
Please always quote using this URN:urn:nbn:de:0296-matheon-8653
- We introduce geodesic finite elements as a conforming way to discretize partial differential equations for functions $v : \Omega \to M$, where $\Omega$ 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 $d=1$ by 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 $S^2$.
Author: | Oliver Sander |
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URN: | urn:nbn:de:0296-matheon-8653 |
Referee: | Peter Deuflhard |
Document Type: | Preprint, Research Center Matheon |
Language: | English |
Date of first Publication: | 2011/06/30 |
Release Date: | 2011/06/30 |
Tag: | Riemannian center of mass; Riemannian trust region method; geodesic finite elements; liquid crystals; manifold; unit sphere |
Institute: | Freie Universität Berlin |
MSC-Classification: | 58-XX GLOBAL ANALYSIS, ANALYSIS ON MANIFOLDS [See also 32Cxx, 32Fxx, 32Wxx, 46-XX, 47Hxx, 53Cxx](For geometric integration theory, see 49Q15) / 58Jxx Partial differential equations on manifolds; differential operators [See also 32Wxx, 35-XX, 53Cxx] / 58J32 Boundary value problems on manifolds |
65-XX NUMERICAL ANALYSIS / 65Nxx Partial differential equations, boundary value problems / 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods | |
Preprint Number: | 802 |