Laplace–Beltrami operator

Summary

In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named after Pierre-Simon Laplace and Eugenio Beltrami. For any twice-differentiable real-valued function f defined on Euclidean space Rn, the Laplace operator (also known as the Laplacian) takes f to the divergence of its gradient vector field, which is the sum of the n pure second derivatives of f with respect to each vector of an orthonormal basis for Rn. Like the Laplacian, the Laplace–Beltrami operator is defined as the divergence of the gradient, and is a linear operator taking functions into functions. The operator can be extended to operate on tensors as the divergence of the covariant derivative. Alternatively, the operator can be generalized to operate on differential forms using the divergence and exterior derivative. The resulting operator is

Official source

https://en.wikipedia.org/wiki/Laplace%E2%80%93Beltrami_operator

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The objective of this PhD thesis is the approximate computation of the solutions of the Spectral Problem associated with the Laplace operator on a compact Riemann surface without boundaries. A Riemann surface can be seen as a gluing of portions of the Hyperbolic Plane made with suitable conditions to obtain a 2 dimensional manifold. The solutions of the Spectral Problem associated with the Laplace operator are to be understood as the eigenfunctions defined on the surface and their corresponding eigenvalues. This work is separated into two parts: the first part describes the method used to approximate the eigenvalues and eigenfunctions, the second focuses on the design of a program to compute these approximations. The approximation method is inspired by the Finite Element Method (FEM), in that it relies on the variational expression of the Spectral Problem and the definition of a finite subspace of functions in which the approximated eigenvalues and eigenfunctions are computed. However, it differs from the FEM in that it removes the euclidian basis of the FEM and is invariant under the isometries of the Hyperbolic Plane. To ful fill this objective, we begin by geodesically triangulating the surface as regularly as possible. This hyperbolic triangulation allows us to de ne the finite subspace of functions by using the concept of barycentric coordinates associated with each vertex of the triangulation (idea introduced by Whitney and taken up by Dodziuk). We then prove that the approximated solutions convergence to the exact ones when the diameter of the triangulation decreases, as well as the order of convergence. The program is a practical application of the theoretical work and allows the computation of the approximated eigenfunctions and eigenvalues.

EPFL2012

Well-Posedness, Regularity, and Convergence Analysis of the Finite Element Approximation of a Generalized Robin Boundary Value Problem

Claudia Maria Colciago, Alfio Quarteroni

In this paper, we propose the mathematical and finite element analysis of a second-order partial differential equation endowed with a generalized Robin boundary condition which involves the Laplace--Beltrami operator by introducing a function space

H^1(\Omega; \Gamma)

H^1(\Omega)

-functions with

H^1(\Gamma)

-traces, where

\Gamma \subseteq \partial \Omega

. Based on a variational method, we prove that the solution of the generalized Robin boundary value problem possesses a better regularity property on the boundary than in the case of the standard Robin problem. We numerically solve generalized Robin problems by means of the finite element method with the aim of validating the theoretical rates of convergence of the error in the norms associated to the space

H^1(\Omega; \Gamma)

Society for Industrial and Applied Mathematics2015

Trace Identities For Commutators, With Applications To The Distribution Of Eigenvalues

Joachim Stubbe

We prove trace identities for commutators of operators, which are used to derive sum rules and sharp universal bounds for the eigenvalues of periodic Schrodinger operators and Schrodinger operators on immersed manifolds. In particular, we prove bounds on the eigenvalue lambda(N+1) in terms of the lower spectrum, bounds on ratios of means of eigenvalues, and universal monotonicity properties of eigenvalue moments, which imply sharp versions of Lieb-Thirring inequalities. In the case of a Schrodinger operator on an immersed manifold of dimension d, we derive a version of Reilly's inequality bounding the eigenvalue lambda(N+1) of the Laplace-Beltrami operator by a universal constant times vertical bar vertical bar h vertical bar N-2(infinity)2/d.

2011

Well-Posedness, Regularity, and Convergence Analysis of the Finite Element Approximation of a Generalized Robin Boundary Value Problem

Claudia Maria Colciago, Alfio Quarteroni

H^1(\Omega; \Gamma)

H^1(\Omega)

-functions with

H^1(\Gamma)

-traces, where

\Gamma \subseteq \partial \Omega

H^1(\Omega; \Gamma)

Society for Industrial and Applied Mathematics2015