We consider the mapping properties of the integral operator arising in nonlocal slender body theory (SBT) for the model geometry of a straight, periodic filament. It is well known that the classical singular SBT integral operator suffers from high wavenumber instabilities, making it unsuitable for approximating the slender body inverse problem, where the fiber velocity is prescribed and the integral operator must be inverted to find the force density along the fiber. Regularizations of the integral operator must therefore be used instead. Here, we consider two regularization methods: spectral truncation and the (Formula presented.) -regularization of Tornberg and Shelley (2004). We compare the mapping properties of these approximations to the underlying partial differential equation (PDE) solution, which for the inverse problem is simply the Stokes Dirichlet problem with data constrained to be constant on cross sections. For the straight-but-periodic fiber with constant radius (Formula presented.), we explicitly calculate the spectrum of the operator mapping fiber velocity to force for both the PDE and the approximations. We prove that the spectrum of the original SBT operator agrees closely with the PDE operator at low wavenumbers but differs at high frequencies, allowing us to define a truncated approximation with a wavenumber cutoff (Formula presented.). For both the truncated and (Formula presented.) -regularized approximations, we obtain rigorous (Formula presented.) -based convergence to the PDE solution as (Formula presented.) : A fiber velocity with (Formula presented.) regularity gives (Formula presented.) convergence, while a fiber velocity with at least (Formula presented.) regularity yields (Formula presented.) convergence. Moreover, we determine the dependence of the (Formula presented.) -regularized error estimate on the regularization parameter (Formula presented.).
All Science Journal Classification (ASJC) codes
- Applied Mathematics
- error analysis
- operator spectrum
- slender body theory