A Singularly Perturbed Boundary Value Problems with Fractional Powers of Elliptic Operators
This work addresses a numerical challenge in solving boundary value problems with small fractional powers of elliptic operators, which is a niche domain-specific problem; the contribution appears incremental.
The paper tackles singularly perturbed boundary value problems involving fractional powers of elliptic operators as the fractional exponent approaches zero, solving them numerically via a pseudo-parabolic time-dependent problem with standard two-level schemes. Numerical results are shown for a model 2D problem.
A boundary value problem for a fractional power $0 < \varepsilon < 1$ of the second-order elliptic operator is considered. The boundary value problem is singularly perturbed when $\varepsilon \rightarrow 0$. It is solved numerically using a time-dependent problem for a pseudo-parabolic equation. For the auxiliary Cauchy problem, the standard two-level schemes with weights are applied. The numerical results are presented for a model two-dimen\-sional boundary value problem with a fractional power of an elliptic operator. Our work focuses on the solution of the boundary value problem with $0 < \varepsilon \ll 1$.