# Numerical Methods for PDEs: In Occasion of Raytcho Lazarov's 70th Birthday

## Joseph Pasciak, Texas A&M University An Analysis of Finite Element Approximation to the Eigenvalues of Problems Involving Fractional Order Differential Operators

### Authors:

• Bangti Jin, Texas A&M University
• Raytcho Lazarov, Texas A&M University
• Joseph Pasciak, Texas A&M University

### Abstract:

In this talk, we consider an eigenvalue problem coming from a boundary value problem involving fractional derivatives. Specifically, we consider the Caputo and Riemann-Liouville fractional differential operators and associated boundary conditions. These boundary value problems will be investigated from a variational point of view. We are interested in the case when the differential operator is of order $$\alpha$$ with $$\alpha \in \left(1, 2\right)$$. These derivatives lead to non-symmetric boundary value problems. The Riemann-Liouville case is somewhat simpler as the underlying variational problem is coercive on a natural subspace of $$H^{\alpha/2} \left(0, 1\right)$$ even though its solutions are less regular. The variational formulation of the Caputo derivative case is more interesting as it leads to a variational problem involving different test and trial spaces. In this case, one is required to prove variational stability on the discrete level as well.

In both cases, the analysis of the eigenvalue problem involves the derivation of "so-called" shift theorems which demonstrate that the solution of the variational problem and its adjoint are more regular, i.e., are in $$H^{\alpha/2+\gamma} \left(0, 1\right)$$ with $$\gamma > 0$$. The regularity pickup enables one to prove that the norm of the solution minus that of the finite element approximation converges with γ dependent rates. This, in turn, can be used to deduce eigenvalue/eigenvector convergence rates. Finally, the results of numerical experiments illustrating the theory will be presented.