A Class of Biharmonic End-strip Problems Arising in Elasticity and Stokes Flow

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Abstract
We consider boundary-value problems for the biharmonic equation in the open rectangle x > 0, −1 < y < 1, with homogeneous boundary conditions on the free edges y=± 1, and data on the end x = 0 of a type arising both in elasticity and in Stokes flow of a viscous fluid, in which either two stresses or two displacements are prescribed. For such “non-canonical” data, coefficients in the eigenfunction expansion can be found only from the solution of infinite sets of linear equations, for which a variety of methods of formulation have been proposed. A drawback of existing methods has been that the resulting equations are unstable with respect to the order of truncation. It is clear from an examination of the spectrum of a typical matrix that ill-conditioning is to be expected. However, a search among a wider class of possible trial functions than hitherto for use in a Galerkin method based on the actual eigenfunctions has led to the choice of a unique set, here termed optimal weighting functions, for which the resulting infinite matrix is diagonally-dominated. This ensures the existence of an inverse, which can be approximated by solving a finite subset of the equations. Computations for a number of representative cases, presented in full in an internal report (Spence, 1978, hereafter referred to as [1]) are summarized here, with emphasis on the rates of decay of the coefficients {cm} in the eigenfunction expansion. Knowledge of these decay rates is essential for a discussion of convergence, parallel to that given by Joseph (1977a, b) and his cò-workers for canonical problems. Asymptotic estimates of the decay rates have also been obtained by use of the solution of the biharmonic equation in a quarter plane. It is found that (i) for smooth continuous data satisfying compatibility conditions at the corners, the decay rates guarantee pointwise convergence. Also examined are (ii) cases of data violating compatibility, (iii) discontinuous data and (iv) discontinuities in derivatives of the data. In these cases sharp estimates of convergence rates are obtained, which guarantee that integrals of the series converge to integrals of the data. The computations show striking confirmation of the theoretical estimates.

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