Ambient Spline Approximation on Manifolds

• Main Author: Odathuparambil, Sonja
• Format: Others
• Language: en
• Published: 2016
• Online Access: https://tuprints.ulb.tu-darmstadt.de/5660/1/Thesis_1.pdf; Odathuparambil, Sonja <http://tuprints.ulb.tu-darmstadt.de/view/person/Odathuparambil=3ASonja=3A=3A.html> (2016): Ambient Spline Approximation on Manifolds.Darmstadt, Technische Universität, [Ph.D. Thesis]

Approximation in $\R^d$ is already well studied while approximation on manifolds still leads to difficulties. The present thesis introduces three approximation methods on compact manifolds. The manifold is considered as a submanifold embedded in $\R^d$ and the problem is extended to some ambient domain of the submanifold. The first method returns $C^k$-approximations, $k \in \N$, of given functions on smooth compact submanifolds. We prove that the method shows optimal approximation behaviour for submanifolds of codimension one. The second and the third method approximate solutions of linear intrinsic PDEs. After extending the problem to some domain around the submanifold boundary conditions are added. The problem is solved by using the Finite Element Method. Numerical test confirm the ideas of the methods.