Anisotropic mesh refinement: The conditioning of Galerkin boundary element matrices and simple preconditioners

I G Graham, W McLean

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14 Citations (Scopus)

Abstract

In this paper we obtain upper and lower bounds on the spectrum of the stiffness matrix arising from a finite element Galerkin approximation (using nodal basis functions) of a bounded, symmetric bilinear form which is elliptic on a Sobolev space of real index m is an element of [- 1, 1]. The key point is that the finite element mesh is required to be neither quasi-uniform nor shape-regular, so that our theory allows anisotropic meshes often used in practice. (However, we assume that the polynomial degree of the elements is fixed.) Our bounds indicate the ill-conditioning which can arise from anisotropic mesh refinement. In addition we obtain spectral bounds for the diagonally scaled stiffness matrix, which indicate the improvement provided by this simple preconditioning. For the special case of boundary integral operators on a two-dimensional screen in R-3, numerical experiments show that our bounds are sharp. We find that diagonal scaling essentially removes the ill-conditioning due to mesh degeneracy, leading to the same asymptotic growth in the condition number as arises for a quasi-uniform mesh refinement. Our results thus generalize earlier work by Bank and Scott [SIAM J. Numer. Anal., 26 (1989), pp. 1383 - 1394] and Ainsworth, McLean, and Tran [SIAM J. Numer. Anal., 36 ( 1999), pp. 1901 - 1932] for the shape-regular case.
Original languageEnglish
Pages (from-to)1487-1513
Number of pages27
JournalSIAM Journal on Numerical Analysis (SINUM)
Volume44
Issue number4
DOIs
Publication statusPublished - 2006

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Anisotropic Mesh
Ill-conditioning
Mesh Refinement
Stiffness matrix
Stiffness Matrix
Conditioning
Preconditioner
Galerkin
Boundary Elements
Spectral Bound
Mesh
Sobolev spaces
Galerkin Approximation
Boundary Integral
Bilinear form
Preconditioning
Condition number
Finite Element Approximation
Degeneracy
Integral Operator

Cite this

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title = "Anisotropic mesh refinement: The conditioning of Galerkin boundary element matrices and simple preconditioners",
abstract = "In this paper we obtain upper and lower bounds on the spectrum of the stiffness matrix arising from a finite element Galerkin approximation (using nodal basis functions) of a bounded, symmetric bilinear form which is elliptic on a Sobolev space of real index m is an element of [- 1, 1]. The key point is that the finite element mesh is required to be neither quasi-uniform nor shape-regular, so that our theory allows anisotropic meshes often used in practice. (However, we assume that the polynomial degree of the elements is fixed.) Our bounds indicate the ill-conditioning which can arise from anisotropic mesh refinement. In addition we obtain spectral bounds for the diagonally scaled stiffness matrix, which indicate the improvement provided by this simple preconditioning. For the special case of boundary integral operators on a two-dimensional screen in R-3, numerical experiments show that our bounds are sharp. We find that diagonal scaling essentially removes the ill-conditioning due to mesh degeneracy, leading to the same asymptotic growth in the condition number as arises for a quasi-uniform mesh refinement. Our results thus generalize earlier work by Bank and Scott [SIAM J. Numer. Anal., 26 (1989), pp. 1383 - 1394] and Ainsworth, McLean, and Tran [SIAM J. Numer. Anal., 36 ( 1999), pp. 1901 - 1932] for the shape-regular case.",
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T1 - Anisotropic mesh refinement: The conditioning of Galerkin boundary element matrices and simple preconditioners

AU - Graham, I G

AU - McLean, W

N1 - ID number: ISI:000241227900007

PY - 2006

Y1 - 2006

N2 - In this paper we obtain upper and lower bounds on the spectrum of the stiffness matrix arising from a finite element Galerkin approximation (using nodal basis functions) of a bounded, symmetric bilinear form which is elliptic on a Sobolev space of real index m is an element of [- 1, 1]. The key point is that the finite element mesh is required to be neither quasi-uniform nor shape-regular, so that our theory allows anisotropic meshes often used in practice. (However, we assume that the polynomial degree of the elements is fixed.) Our bounds indicate the ill-conditioning which can arise from anisotropic mesh refinement. In addition we obtain spectral bounds for the diagonally scaled stiffness matrix, which indicate the improvement provided by this simple preconditioning. For the special case of boundary integral operators on a two-dimensional screen in R-3, numerical experiments show that our bounds are sharp. We find that diagonal scaling essentially removes the ill-conditioning due to mesh degeneracy, leading to the same asymptotic growth in the condition number as arises for a quasi-uniform mesh refinement. Our results thus generalize earlier work by Bank and Scott [SIAM J. Numer. Anal., 26 (1989), pp. 1383 - 1394] and Ainsworth, McLean, and Tran [SIAM J. Numer. Anal., 36 ( 1999), pp. 1901 - 1932] for the shape-regular case.

AB - In this paper we obtain upper and lower bounds on the spectrum of the stiffness matrix arising from a finite element Galerkin approximation (using nodal basis functions) of a bounded, symmetric bilinear form which is elliptic on a Sobolev space of real index m is an element of [- 1, 1]. The key point is that the finite element mesh is required to be neither quasi-uniform nor shape-regular, so that our theory allows anisotropic meshes often used in practice. (However, we assume that the polynomial degree of the elements is fixed.) Our bounds indicate the ill-conditioning which can arise from anisotropic mesh refinement. In addition we obtain spectral bounds for the diagonally scaled stiffness matrix, which indicate the improvement provided by this simple preconditioning. For the special case of boundary integral operators on a two-dimensional screen in R-3, numerical experiments show that our bounds are sharp. We find that diagonal scaling essentially removes the ill-conditioning due to mesh degeneracy, leading to the same asymptotic growth in the condition number as arises for a quasi-uniform mesh refinement. Our results thus generalize earlier work by Bank and Scott [SIAM J. Numer. Anal., 26 (1989), pp. 1383 - 1394] and Ainsworth, McLean, and Tran [SIAM J. Numer. Anal., 36 ( 1999), pp. 1901 - 1932] for the shape-regular case.

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DO - 10.1137/040621247

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JF - SIAM Journal on Numerical Analysis (SINUM)

SN - 0036-1429

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