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On integral equation methods for the first Dirichlet problem of the biharmonic and modified biharmonic equations in nonsmooth domains

Helsing, Johan LU and Jiang, Shidong (2018) In SIAM Journal on Scientific Computing 40(4). p.2609-2630
Abstract

Despite important applications in unsteady Stokes flow, a Fredholm second kind integral equation formulation modeling the first Dirichlet problem of the modified biharmonic equation in the plane has been derived only recently. Furthermore, this formulation becomes very ill-conditioned when the boundary is not smooth, say, having corners. The present work demonstrates numerically that a method called recursively compressed inverse preconditioning (RCIP) can be effective when dealing with this geometrically induced ill-conditioning in the context of Nystr\"om discretization. The RCIP method not only reduces the number of iterations needed in iterative solvers but also improves the achievable accuracy in the solution. Adaptive mesh... (More)

Despite important applications in unsteady Stokes flow, a Fredholm second kind integral equation formulation modeling the first Dirichlet problem of the modified biharmonic equation in the plane has been derived only recently. Furthermore, this formulation becomes very ill-conditioned when the boundary is not smooth, say, having corners. The present work demonstrates numerically that a method called recursively compressed inverse preconditioning (RCIP) can be effective when dealing with this geometrically induced ill-conditioning in the context of Nystr\"om discretization. The RCIP method not only reduces the number of iterations needed in iterative solvers but also improves the achievable accuracy in the solution. Adaptive mesh refinement is only used in the construction of a compressed inverse preconditioner, leading to an optimal number of unknowns in the linear system in the solve phase.

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type
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publication status
published
subject
keywords
Biharmonic equation, Modified biharmonic equation, RCIP method, Second kind integral equation
in
SIAM Journal on Scientific Computing
volume
40
issue
4
pages
2609 - 2630
publisher
Society for Industrial and Applied Mathematics
external identifiers
  • scopus:85053775840
ISSN
1064-8275
DOI
10.1137/17M1162238
language
English
LU publication?
yes
id
ad5735e2-de77-48d7-8663-5896b85bfe3d
date added to LUP
2018-10-23 12:43:22
date last changed
2021-10-10 03:02:59
@article{ad5735e2-de77-48d7-8663-5896b85bfe3d,
  abstract     = {<p>Despite important applications in unsteady Stokes flow, a Fredholm second kind integral equation formulation modeling the first Dirichlet problem of the modified biharmonic equation in the plane has been derived only recently. Furthermore, this formulation becomes very ill-conditioned when the boundary is not smooth, say, having corners. The present work demonstrates numerically that a method called recursively compressed inverse preconditioning (RCIP) can be effective when dealing with this geometrically induced ill-conditioning in the context of Nystr\"om discretization. The RCIP method not only reduces the number of iterations needed in iterative solvers but also improves the achievable accuracy in the solution. Adaptive mesh refinement is only used in the construction of a compressed inverse preconditioner, leading to an optimal number of unknowns in the linear system in the solve phase.</p>},
  author       = {Helsing, Johan and Jiang, Shidong},
  issn         = {1064-8275},
  language     = {eng},
  number       = {4},
  pages        = {2609--2630},
  publisher    = {Society for Industrial and Applied Mathematics},
  series       = {SIAM Journal on Scientific Computing},
  title        = {On integral equation methods for the first Dirichlet problem of the biharmonic and modified biharmonic equations in nonsmooth domains},
  url          = {http://dx.doi.org/10.1137/17M1162238},
  doi          = {10.1137/17M1162238},
  volume       = {40},
  year         = {2018},
}