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Concept# Laplace operator

Summary

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the nabla operator), or \Delta. In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian Δf (p) of a function f at a point p measures by how much the average value of f over small spheres or balls centered at p deviates from f (p).
The Laplace operator is named after the French mathematician Pierre-Simon de Laplace (1749–1827), who first applied the operator to the study of celestial mechanics: the Laplacian of the gravitational potential due to

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We show that isogeometric Galerkin discretizations of eigenvalue problems related to the Laplace operator subject to any standard type of homogeneous boundary conditions have no outliers in certain optimal spline subspaces. Roughly speaking, these optimal subspaces are obtained from the full spline space defined on certain uniform knot sequences by imposing specific additional boundary conditions. The spline subspaces of interest have been introduced in the literature some years ago when proving their optimality with respect to Kolmogorov n-widths in L2-norm for some function classes. The eigenfunctions of the Laplacian - with any standard type of homogeneous boundary conditions - belong to such classes. Here we complete the analysis of the approximation properties of these optimal spline subspaces. In particular, we provide explicit L2 and H1 error estimates with full approximation order for Ritz projectors in the univariate and in the multivariate tensor-product setting. Besides their intrinsic interest, these estimates imply that, for a fixed number of degrees of freedom, all the eigenfunctions and the corresponding eigenvalues are well approximated, without loss of accuracy in the whole spectrum when compared to the full spline space. Moreover, there are no spurious values in the approximated spectrum. In other words, the considered subspaces provide accurate outlier-free discretizations in the univariate and in the multivariate tensor-product case. This main contribution is complemented by an explicit construction of B-spline-like bases for the considered spline subspaces. The role of such spaces as accurate discretization spaces for addressing general problems with non-homogeneous boundary behavior is discussed as well. c 2021 Elsevier B.V. All rights reserved.

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.

We complete the picture of sharp eigenvalue estimates for the -Laplacian on a compact manifold by providing sharp estimates on the first nonzero eigenvalue of the nonlinear operator when the Ricci curvature is bounded from below by a negative constant. We assume that the boundary of the manifold is convex, and put Neumann boundary conditions on it. The proof is based on a refined gradient comparison technique and a careful analysis of the underlying model spaces.