ON THE LOW FREQUENCY ASYMPTOTICS OF THE EXTERIOR 2-D DIRICHLET PROBLEM IN DYNAMIC ELASTICITY

ABSTRACT Here we present the low frequency behaviour of the solution of the exterior 2-D Dirichlet problem of elasticity and show its convergence to the corresponding stationary field with the order ***(lnω) −1 in compact domains where ω denotes the frequency. The analysis is based on a strongly elliptic Fredholm integral equation of the first kind on the boundary which provides the complete asymptotics as well as an asymptotic expansion in powers of (lnω) −1 . The first kind integral equation is a pseudodifferential equation of order −1 and, hence, ill posed in L 2 . We also present asymptotic error and stability estimates for spline Galerkin and spline collocation approximations which define regularizations by the projection onto the finite dimensional spline spaces. Here this regularization is asymptotically of the same accuracy and stability as the Tikhonov regularization in combination with the spline Galerkin approximation and optimal choice of the Tikhonov parameter.