An adaptive multimesh rational approximation scheme for the spectral fractional Laplacian

TL;DR

This work develops an adaptive multimesh rational approximation framework for the spectral fractional Laplacian $(-\Delta)^s$ with Dirichlet boundary conditions. It combines the Bonito–Pasciak rational approximation of $\lambda^{-s}$ with parametric nonfractional PDEs solved on per-parameter meshes, assembling the final solution on a union mesh. A Bank–Weiser–style a posteriori estimator guides an adaptive multimesh refinement, and two global estimators (triangle-inequality and union-mesh) are analyzed. Numerical experiments demonstrate faster convergence and up to 10x reduction in cumulative degrees of freedom compared to single-mesh adaptivity, with robust performance across test cases and potential for further gains via anisotropic refinement.

Abstract

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 $λ\mapsto λ^{-s}$ 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.

Figures (6)

• Figure 1: Test case I (multimesh setting; $s \in \{0.3,\, 0.5,\, 0.7\}$): evolution of global error estimates $\eta^m$, $\widetilde{\eta}^m$ and the corresponding effectivity indices $\Theta^m$, $\widetilde{\Theta}^m$.
• Figure 2: Test case II: evolution of global error estimates in the single-mesh setting ($\widetilde{\eta}^m$ defined by \ref{['eq:union_mesh_fractional_estimator1']}) and in the multimesh setting ($\eta^m$ and $\widetilde{\eta}^m$ defined by \ref{['eq:triangular_inequality_fractional_estimator']} and \ref{['eq:union_mesh_fractional_estimate_def']}, respectively) for $s \in \{0.3,\, 0.5,\, 0.7\}$.
• Figure 3: Test case II: cumulative costs (as defined in \ref{['eq:cumulative_cost']}) plotted against the global error estimates $\widetilde{\eta}^m$ at each iteration of adaptive loop in the single-mesh (circular markers) and multimesh (square markers) settings.
• Figure 4: Test case II (multimesh setting; $s=0.3$ (top row), $s=0.5$ (middle row), and $s=0.7$ (bottom row)): the number of refinements for each mesh $\mathcal{T}_l$ ($l = 1,\ldots,N$) (right column) and the corresponding weighted error estimates $a_l \eta_l^m$ (see \ref{['eq:global_parametric_estimator']}) (left column). In the left column, different shades represent different iterations $m$ of the adaptive loop.
• Figure 5: Test case II (multimesh setting; $s=0.3$): the initial coarse mesh $\mathcal{T}^0$; finite element meshes $\mathcal{T}_l^m$ generated by Algorithm \ref{['alg:algorithm_outline']} ($m=19$; $l=82,\, 109,\, 122,\, 127$); the union mesh $\widetilde{\mathcal{T}}^{19}$.