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 $λ\mapsto λ^{-s}$ 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.

💡 Research Summary

The paper addresses the computational challenges associated with solving the homogeneous Dirichlet problem for the spectral fractional Laplacian ((-\Delta)^s) on bounded domains. Direct finite‑element discretization of the fractional operator leads to dense matrices and prohibitive costs, especially in three dimensions or for fine meshes. A popular remedy is to approximate the function (\lambda^{-s}) by a rational function and rewrite the fractional problem as a sum of independent, non‑fractional reaction‑diffusion equations. Existing rational‑approximation schemes, however, use a single finite‑element mesh for all auxiliary PDEs, which is inefficient because the diffusion coefficients (b_\ell) in the auxiliary problems can vary by several orders of magnitude.

The authors propose an adaptive multimesh rational approximation scheme. First, they adopt the Bonito‑Pasciak (BP) rational approximation, which expresses (\lambda^{-s}) as a weighted sum of terms of the form (\frac{a_\ell}{c_\ell + b_\ell \lambda}). The weights (a_\ell), diffusion coefficients (b_\ell), and reaction coefficients (c_\ell) are explicitly given and depend on a fineness parameter (\kappa). The approximation error decays exponentially in (\kappa) and can be bounded by a computable quantity (\varepsilon_s(\kappa)) that involves only the lowest eigenvalue (\lambda_0) of the Laplacian.

For each term (\ell) the corresponding auxiliary PDE \