A kernel-free boundary integral method for elliptic boundary value problems

Ying, Wenjun; Henriquez, Craig S.

doi:10.1016/j.jcp.2007.08.021

Cited by 53 publications

(35 citation statements)

References 70 publications

(150 reference statements)

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“…Thus, if the discrete system corresponds to an integral equation of the second kind, then GMRES can converge quickly. (We refer the readers to [49] for the relation between an augmented approach and the boundary integral method.) One such an example is to solve a Poisson equation on an irregular domain with different boundary conditions.…”

Section: Features Of the Schur Complement Systems And Challenges In Gmentioning

confidence: 99%

Effective matrix-free preconditioning for the augmented immersed interface method

Xia

2015

Journal of Computational Physics

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Please cite this article in press as: J. Xia et al., Effective matrix-free preconditioning for the augmented immersed interface method, J. Comput. Phys. (2015), http://dx. AbstractWe present effective and efficient matrix-free preconditioning techniques for the augmented immersed interface method (AIIM). AIIM has been developed recently and is shown to be very effective for interface problems and problems on irregular domains. GMRES is often used to solve for the augmented variable(s) associated with a Schur complement A in AIIM that is defined along the interface or the irregular boundary. The efficiency of AIIM relies on how quickly the system for A can be solved. For some applications, there are substantial difficulties involved, such as the slow convergence of GMRES (particularly for free boundary and moving interface problems), and the inconvenience in finding a preconditioner (due to the situation that only the products of A and vectors are available). Here, we propose matrix-free structured preconditioning techniques for AIIM via adaptive randomized sampling, using only the products of A and vectors to construct a hierarchically semiseparable matrix approximation to A. Several improvements over existing schemes are shown so as to enhance the efficiency and also avoid potential instability. The significance of the preconditioners includes: (1) they do not require the entries of A or the multiplication of A T with vectors; (2) constructing the preconditioners needs only O(log N) matrix-vector products and O(N) storage, where N is the size of A; (3) applying the preconditioners needs only O(N) flops;(4) they are very flexible and do not require any a priori knowledge of the structure of A. The preconditioners are observed to significantly accelerate the convergence of GMRES, with heuristical justifications of the effectiveness. Comprehensive tests on several important applications are provided, such as Navier-Stokes equations on irregular domains with traction boundary conditions, interface problems in incompressible flows, mixed boundary problems, and free boundary problems. The preconditioning techniques are also useful for several other problems and methods.

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Section: Features Of the Schur Complement Systems And Challenges In Gmentioning

confidence: 99%

Effective matrix-free preconditioning for the augmented immersed interface method

Xia

2015

Journal of Computational Physics

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“…On the other hand, we may have less knowledge about the condition number of the Schur-complement system and how to apply pre-conditioning techniques. Recently, Ying and Henriquez [165] developed the augmented method (they labeled it as a kernel-free boundary integral method) for elliptic boundary value problems on irregular domains. The analysis is based on operator theory, for example, [69,71].…”

Section: The Augmented Algorithm For Navier-stokes Equations On Irregmentioning

confidence: 99%

“…The analysis is based on operator theory, for example, [69,71]. We believe that the analysis in [165] can be applied to augmented methods for different problems.…”

Section: The Augmented Algorithm For Navier-stokes Equations On Irregmentioning

confidence: 99%

An Introduction to the Immersed Boundary and the Immersed Interface Methods

Dillon¹,

Li²

2009

Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore

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This paper attempts to give a brief survey or tutorial on the immersed boundary (IB) method and the immersed interface method (IIM). The immersed boundary method was originally introduced by Peskin for studying flow patterns around heart valves and for studying blood flow in the heart [118] and has since been applied to many other problems, particularly, in biophysics. The IB method is a mathematical modeling framework as well as a numerical method. The original motivation of the immersed interface method [76,80] was to improve the accuracy of the IB method, at least from first-order accuracy to second; and deal with discontinuous coefficients in the governing equations. The IIM is a sharp interface method based on Cartesian grids or triangulation. The IIM makes use of jump conditions across interfaces so that the finite difference/element discretization can be accurate. Some new developments in the immersed interface method include the augmented 1 Interface Problems and Methods in Biological and Physical Flows Downloaded from www.worldscientific.com by Mr Yaosheng Liang on 03/20/15. For personal use only. 2 R. H. Dillon and Z. Li immersed interface method; immersed finite element method; and singularity removal techniques. The immersed boundary and the interface methods have been coupled with different evolution schemes for free boundary and moving interface problems.Keywords: Immersed boundary method, immersed interface method, interface problem, irregular domain, discontinuous coefficient, singular source, delta function, discontinuous or non-smooth solution, augmented method, immersed finite element method, source singularity removal.

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“…Another recent work in this area is a class of kernelfree boundary integral (KFBI) methods for solving elliptic BVPs, presented in [25].…”

Section: Introductionmentioning

confidence: 99%

Numerical Method for Solving Matrix Coefficient Elliptic Equation on Irregular Domains with Sharp-Edged Boundaries

Wang

Shi

2013

Advances in Numerical Analysis

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We present a new second-order accurate numerical method for solving matrix coefficient elliptic equation on irregular domains with sharp-edged boundaries. Nontraditional finite element method with non-body-fitting grids is implemented on a fictitious domain in which the irregular domains are embedded. First we set the function and coefficient in the fictitious part, and the nonsmooth boundary is then treated as an interface. The emphasis is on the construction of jump conditions on the interface; a special position for the ghost point is chosen so that the method is more accurate. The test function basis is chosen to be the standard finite element basis independent of the interface, and the solution basis is chosen to be piecewise linear satisfying the jump conditions across the interface. This is an efficient method for dealing with elliptic equations in irregular domains with non-smooth boundaries, and it is able to treat the general case of matrix coefficient. The complexity and computational expense in mesh generation is highly decreased, especially for moving boundaries, while robustness, efficiency, and accuracy are promised. Extensive numerical experiments indicate that this method is second-order accurate in the ∞ norm in both two and three dimensions and numerically very stable.

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A kernel-free boundary integral method for elliptic boundary value problems

Cited by 53 publications

References 70 publications

Effective matrix-free preconditioning for the augmented immersed interface method

Effective matrix-free preconditioning for the augmented immersed interface method

An Introduction to the Immersed Boundary and the Immersed Interface Methods

Numerical Method for Solving Matrix Coefficient Elliptic Equation on Irregular Domains with Sharp-Edged Boundaries

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