Third and fifth order quadrature rules for investigating harmonic and biharmonic boundary integral operators

Søren Christiansen (

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Three integral operators related to harmonic and biharmonic boundary integral equations are replaced by matrices derived by a quadrature method with convergence order for the harmonic operator but for the biharmonic operators. The fifth order convergence is obtained by an expedient change of the diagonal elements of the matrix; without this change the convergence order is only three.

For a circular boundary curve the radii, which correspond to zeros of the eigenvalues, we will denote xxx. This xxx is of particular interest in connection with "diagnostic" investigations of the operator. For the harmonic operator the critical values of the operator and of the matrix are xxx (even when the convergence order is only three). For the two biharmonic operators the critical values of the matrices converge to those of the operator with order xxx. Therefore a matrix derived by this quadrature method is an accurate tool for detecting critical cases of the corresponding integral operator.

The results have been obtained by an extensive use of the computer algebra system Maple V, which has been a crucial tool.

Keywords: Integral operators, Matrices, Quadrature methods, Order of convergence, Eigenvalues, Zeros.

Last modified Jan. 20, 1999

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