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Fast Poisson Fast Helmholtz and Fast Linear Elastostatic Solvers

Solvers

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Fast Poisson, Fast Helmholtz and Fast Linear Elastostatic Solvers on Rectangular Parallelepipeds.pdf

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LBNL·43565

Preprint

ERNEST

ORLANDO

LAWRENCE

BERKELEY

NATIONAL

LABORATORY

Fast

Poisson,

Fast

Helmholtz and

Fast

Linear

Elastostatic

Solvers

.'!,'

on Rectangular Parallelepipeds

Andreas

Wiegmann

Computing

Sciences

Directorate

Mathematics Department

June

1999

submitted

for

publication

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DISCLAIMER

This document was prepared

account of work sponsored

the

United States Government. While this document

believed

contain

correct information, neither the United States Government nor

any

agency thereof, nor The Regents of the University

California, nor

any

of their employees, makes

any

warranty, express or implied, or assumes

any legal responsibility for the accuracy, completeness, or usefulness

any information, apparatus, product, or process disclosed, or

represents that its use would not infringe privately

owped rights.

Reference herein

any specific commercial product, process,

service

its trade name, trademark, manufacturer, or otherwise,

does

not necessarily constitute or imply its endorsement,

recommendation,

or favoring

the United States Government or any agency thereof,

The Regents of the University of California. The views and opinions

authors expressed herein

not necessarily state or reflect those of

the

United States Government or any agency thereof, or The Regents of

the

University of California.

Ernest

Orlando Lawrence Berkeley National Laboratory

is an equal opportunity employer.

LBNL-43565

Fast Poisson, Fast Helmholtz and Fast Linear Elastostatic

Solvers on Rectangular Parallelepipeds

Andreas Wiegmann

Computing Sciences Directorate

Department

Mathematics

Ernest

Orlando Lawrence Berkeley National Laboratory

University

California

Berkeley, California 94720

June 1999

This work was supported by the Office

Science, Office

Advanced Scientific Computing Research, .

Mathematical, Information and Computational Science Division, Applied Mathematical Sciences Subprogram,

the U.S. Department

Energy under Contract No. DE-AC03-76SF00098.

* zocycled

paper

Abstract

FFT-based

fast

Poisson

and

fast Helmholtz solvers on rectangular parallelepipeds for periodic boundary

conditions in one-, two

and

three space dimensions can also be used to solve Dirichlet

and

Neumann boundary

value problems. For non-zero boundary conditions, this

the special, grid-aligned case of jump corrections

used in

the

Explicit

Jump

Immersed Interface method.

Fast elastostatic solvers for periodic boundary conditions in two and three dimensions can also be based

the

FFT.

From

the

periodic solvers

derive fast solvers for

the

new "normal" boundary conditions

and essential boundary conditions on rectangular parallelepipeds.

The

periodic case allows a simple proof

of existence

and

uniqueness of

the

solutions

the

discretization of normal boundary conditions. Numerical

examples demonstrate

the

efficiency of

the

fast elastostatic solvers for non-periodic boundary conditions.

More importantly,

the

fast solvers on rectangular parallelepipeds can be used together with

the

Immersed

Interface Method

solve problems on non-rectangular domains with general boundary conditions. Details

of this are reported in

the

preprint The Explicit Jump Immersed Interface Method for 2D Linear Elastostatics

the

author.

Keywords:

Fast Helmholtz solver, fast Poisson solver, fast elastostatic solver, boundary conditions.

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