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

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Søren Christiansen * (sc@imm.dtu.dk) *

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**Abstract:**
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