Second kind integral equations for the first kind Dirichlet problem of the biharmonic equation in three dimensions

Jiang, Shidong; Ren, Bo; Tsuji, Paul H.; Ying, Lexing

doi:10.1016/j.jcp.2011.06.015

Cited by 11 publications

(12 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [18], an SKIE is derived for (15)-- (17). The representation for u and the consequent SKIE system are formally identical to (8)--(11), but the kernels G i and G ij now become (19) where the functions \scrC k (k = 0, . .…”

Section: The First Dirichlet Problem Of the Biharmonic Equationmentioning

confidence: 99%

“…The modified biharmonic SKIE system (9) with (19) is discretized using that same Nystr\" om scheme but supplemented with a product integration scheme [14, section 6.1] which is activated when y is close to x. This requires that each kernel is split into a smooth part and a logarithmically singular part,…”

Section: The First Dirichlet Problem Of the Biharmonic Equationmentioning

confidence: 99%

“…where the factor G ij are smooth in the sense that, for a fixed x, their dependence on y is well approximated panelwise by polynomials of degree n gl -1. Without ambition of being exhaustive, the rest of this section summarizes some results needed to implement the discretizations of (9) with (12) and (19).…”

Section: The First Dirichlet Problem Of the Biharmonic Equationmentioning

confidence: 99%

“…For the kernels in (19), we need G (L) ij of (22). These functions can be constructed from splittings (25) \scrC…”

Section: The First Dirichlet Problem Of the Biharmonic Equationmentioning

confidence: 99%

“…Stable splitting for the modified biharmonic problem. For the modified biharmonic problem (9) with (19) and (36), operators with logarithmically singular kernels are not smooth and thus need to be kept in K \star . It is straightforward to split those operators into two parts---K (S) and K (L) , with K (S) containing smooth terms, which would lead to large cancellation error unless they are included in K \circ , and K (L) containing the rest.…”

Section: 42mentioning

confidence: 99%

See 4 more Smart Citations

On Integral Equation Methods for the First Dirichlet Problem of the Biharmonic and Modified Biharmonic Equations in NonSmooth Domains

Helsing

Jiang

2018

SIAM J. Sci. Comput.

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: The First Dirichlet Problem Of the Biharmonic Equationmentioning

confidence: 99%

Section: The First Dirichlet Problem Of the Biharmonic Equationmentioning

confidence: 99%

Section: The First Dirichlet Problem Of the Biharmonic Equationmentioning

confidence: 99%

“…For the kernels in (19), we need G (L) ij of (22). These functions can be constructed from splittings (25) \scrC…”

Section: The First Dirichlet Problem Of the Biharmonic Equationmentioning

confidence: 99%

Section: 42mentioning

confidence: 99%

See 3 more Smart Citations

On Integral Equation Methods for the First Dirichlet Problem of the Biharmonic and Modified Biharmonic Equations in NonSmooth Domains

Helsing

Jiang

2018

SIAM J. Sci. Comput.

Self Cite

View full text Add to dashboard Cite

show abstract

On the Solution of the Stokes Equation on Regions with Corners

Rachh

Serkh

2020

Comm Pure Appl Math

View full text Add to dashboard Cite

The detailed behavior of solutions to Stokes equations on regions with corners has been historically difficult to characterize. The solutions to Stokes equations on regions with corners are known to develop singularities in the vicinity of corners; in particular, the solutions are known to have infinite oscillations along almost every ray that meet at the corner. While the nature of singularities for the differential equation have been analyzed in great detail, very little is known about the nature of singularities for the corresponding integral equations. In this paper, we observe that, when the Stokes equation is formulated as a boundary integral equation, the solutions are representable by rapidly convergent series of the form ∑j()cjtμjsin()βjlog()t+djtμjcos()βjlog()t, where t is the distance from the corner and the parameters μj, βj are real, and are determined via an explicit formula depending on the angle at the corner. In addition to being analytically perspicuous, these representations lend themselves to the construction of highly accurate and efficient numerical discretizations, significantly reducing the number of degrees of freedom required for the solution of the corresponding integral equations. The results are illustrated by several numerical examples. © 2020 Wiley Periodicals LLC

show abstract

An iterative regularizing method for an incomplete boundary data problem for the biharmonic equation

Chapko

Johansson

2018

Z Angew Math Mech

View full text Add to dashboard Cite

An incomplete boundary data problem for the biharmonic equation is considered, where the displacement is known throughout the boundary of the solution domain whilst the normal derivative and bending moment are specified on only a portion of the boundary. For this inverse ill‐posed problem an iterative regularizing method is proposed for the stable data reconstruction on the underspecified boundary part. Convergence is proven by showing that the method can be written as a Landweber‐type procedure for an operator formulation of the incomplete data problem. This reformulation renders a stopping rule, the discrepancy principle, for terminating the iterations in the case of noisy data. Uniqueness of a solution to the considered problem is also shown.

show abstract

Second kind integral equations for the first kind Dirichlet problem of the biharmonic equation in three dimensions

Cited by 11 publications

References 29 publications

On Integral Equation Methods for the First Dirichlet Problem of the Biharmonic and Modified Biharmonic Equations in NonSmooth Domains

On Integral Equation Methods for the First Dirichlet Problem of the Biharmonic and Modified Biharmonic Equations in NonSmooth Domains

On the Solution of the Stokes Equation on Regions with Corners

An iterative regularizing method for an incomplete boundary data problem for the biharmonic equation

Contact Info

Product

Resources

About