DeepAI AI Chat
Log In Sign Up

Domain Decomposition for the Closest Point Method

by   Ian May, et al.
Memorial University of Newfoundland
Simon Fraser University

The discretization of elliptic PDEs leads to large coupled systems of equations. Domain decomposition methods (DDMs) are one approach to the solution of these systems, and can split the problem in a way that allows for parallel computing. Herein, we extend two DDMs to elliptic PDEs posed intrinsic to surfaces as discretized by the Closest Point Method (CPM) SJR:CPM,CBM:ICPM. We consider the positive Helmholtz equation (c-Δ_S)u = f, where c∈R^+ is a constant and Δ_S is the Laplace-Beltrami operator associated with the surface S⊂R^d. The evolution of diffusion equations by implicit time-stepping schemes and Laplace-Beltrami eigenvalue problems CBM:Eig both give rise to equations of this form. The creation of efficient, parallel, solvers for this equation would ease the investigation of reaction-diffusion equations on surfaces CBM:RDonPC, and speed up shape classification Reuter:ShapeDNA, to name a couple applications.


page 1

page 2

page 3

page 4


A closest point method library for PDEs on surfaces with parallel domain decomposition solvers and preconditioners

The DD-CPM software library provides a set of tools for the discretizati...

Overlapping Schwarz methods with GenEO coarse spaces for indefinite and non-self-adjoint problems

GenEO ('Generalised Eigenvalue problems on the Overlap') is a method for...

The fundamental solution of a 1D evolution equation with a sign changing diffusion coefficient

In this work we investigate a 1D evolution equation involving a divergen...

Additive Schwarz solvers and preconditioners for the closest point method

The discretization of surface intrinsic elliptic partial differential eq...

A Convergence Analysis of the Parallel Schwarz Solution of the Continuous Closest Point Method

The discretization of surface intrinsic PDEs has challenges that one mig...

Overlapping Domain Decomposition Preconditioner for Integral Equations

The discretization of certain integral equations, e.g., the first-kind F...

Scheduled Relaxation Jacobi schemes for non-elliptic partial differential equations

The Scheduled Relaxation Jacobi (SRJ) method is a linear solver algorith...