Bootstrap Multigrid for the Laplace-Beltrami Eigenvalue Problem

This paper introduces bootstrap two-grid and multigrid finite element approximations to the Laplace-Beltrami (surface Laplacian) eigen-problem on a closed surface. The proposed multigrid method is suitable for recovering eigenvalues having large multiplicity, computing interior eigenvalues, and approximating the shifted indefinite eigen-problem. Convergence analysis is carried out for a simplified two-grid algorithm and numerical experiments are presented to illustrate the basic components and ideas behind the overall bootstrap multigrid approach

Finite Element Approximation of Eigenvalues and Eigenfunctions of the Laplace-Beltrami Operator

The surface finite element method is an important tool for discretizing and solving elliptic partial differential equations on surfaces. Recently the surface finite element method has been used for computing approximate eigenvalues and eigenfunctions of the Laplace-Beltrami operator, but no theoretical analysis exists to offer computational guidance. In this dissertation we develop approximations of the eigenvalues and eigenfunctions of the Laplace-Beltrami operator using the surface finite element method. We develop a priori estimates for the eigenvalues and eigenfunctions of the Laplace-Beltrami operator. We then use these a priori estimates to develop and analyze an optimal adaptive method for approximating eigenfunctions of the Laplace-Beltrami operator

Numerical Approximation of Gaussian random fields on Closed Surfaces

We consider the numerical approximation of Gaussian random fields on closed surfaces defined as the solution to a fractional stochastic partial differential equation (SPDE) with additive white noise. The SPDE involves two parameters controlling the smoothness and the correlation length of the Gaussian random field. The proposed numerical method relies on the Balakrishnan integral representation of the solution and does not require the approximation of eigenpairs. Rather, it consists of a sinc quadrature coupled with a standard surface finite element method. We provide a complete error analysis of the method and illustrate its performances by several numerical experiments.Comment: 33 pages, 2 figures, 5 table

Galerkin--Chebyshev approximation of Gaussian random fields on compact Riemannian manifolds

A new numerical approximation method for a class of Gaussian random fields on compact Riemannian manifolds is introduced. This class of random fields is characterized by the Laplace--Beltrami operator on the manifold. A Galerkin approximation is combined with a polynomial approximation using Chebyshev series. This so-called Galerkin--Chebyshev approximation scheme yields efficient and generic sampling algorithms for Gaussian random fields on manifolds. Strong and weak orders of convergence for the Galerkin approximation and strong convergence orders for the Galerkin--Chebyshev approximation are shown and confirmed through numerical experiments.Comment: Version submitted to a peer-reviewed journal. Changes: fixed residual typos, new outline. 33 pages, 5 figure