Pollution-free approximation techniques for fractional integro-differential equations

Computing eigenvalues of an operator with a non-trivial essential spectrum is problematic. Galerkin methods are in general unreliable and may include a large number of spurious eigenvalues.

This phenomenon called spectral pollution is a major difficulty in numerical analysis and a well-reported difficulty in many different areas of physics.

Pollution-free approximation techniques have been developed for differential equations but not for integro-differential equations.In the project, we will for the first time develop an approximation technique for fractional integro-differential equations, which avoid spurious eigenvalues in gaps between parts of the essential spectrum.

We base the numerical analysis on the corresponding operator function and suggest perturbing the operator coefficients by particular finite rank operators.

The first aim is to prove that standard Galerkin methods can be applied to the perturbed operator function without any spectral pollution and control precisely how the eigenvalues are perturbed.To make full use of the new pollution-free approximation technique, we aim to derive residual estimators forhigh-order finite element methods.

The new algorithms will together translate into pollution-free and highlyaccurate approximations of the spectra of fractional integro-differential equations with a non-trivial essential spectrum.