21 research outputs found
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
A reduced conjugate gradient basis method for fractional diffusion
This work is on a fast and accurate reduced basis method for solving
discretized fractional elliptic partial differential equations (PDEs) of the
form by rational approximation. A direct computation of the
action of such an approximation would require solving multiple (2030)
large-scale sparse linear systems. Our method constructs the reduced basis
using the first few directions obtained from the preconditioned conjugate
gradient method applied to one of the linear systems. As shown in the theory
and experiments, only a small number of directions (510) are needed to
approximately solve all large-scale systems on the reduced basis subspace. This
reduces the computational cost dramatically because: (1) We only use one of the
large-scale problems to construct the basis; and (2) all large-scale problems
restricted to the subspace have much smaller sizes. We test our algorithms for
fractional PDEs on a 3d Euclidean domain, a 2d surface, and random
combinatorial graphs. We also use a novel approach to construct the rational
approximation for the fractional power function by the orthogonal greedy
algorithm (OGA)
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