An adaptive multimesh rational approximation scheme for the spectral fractional Laplacian

We propose 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 family of finite element discretizations 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 existing single-mesh approaches that employ the same finite element mesh across 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 demonstrate that our adaptive multimesh approach achieves faster convergence rates than uniform mesh refinement and yields significant reductions in computational costs -- both in terms of the overall number of degrees of freedom and the actual runtime -- when compared to the corresponding adaptive algorithm in a single-mesh setting.