An adaptive multimesh rational approximation scheme for the spectral fractional Laplacian

The paper presents a novel multimesh rational approximation scheme for the numerical solution of the (homogeneous) Dirichlet problem for the spectral fractional Laplacian. The scheme combines a rational approximation of the function $\lambda \mapsto \lambda^{-s}$ with a set of finite element approximations of parameter-dependent non-fractional partial differential equations (PDEs). The key idea that underpins the proposed scheme is that each parametric PDE is numerically solved on an individually tailored finite element mesh. This is in contrast to the existing single-mesh approach, where the same finite element mesh is employed for solving all parametric PDEs. We develop an a posteriori error estimation strategy for the proposed rational approximation scheme and design an adaptive multimesh refinement algorithm. Numerical experiments show improvements in convergence rates compared to the rates for uniform mesh refinement and up to 10 times reduction in computational costs compared to the corresponding adaptive algorithm in the single-mesh setting.