A New $\mathcal{L}1$-TFPM Scheme for the Singularly Perturbed Subdiffusion Equations (CSIAM Transactions on Applied Mathematics, 2025)
Since the memory effect is taken into account, the singularly perturbed subdiffusion equation can better describe the diffusion phenomenon with small diffusioncoefficients. However, near the boundary configured with non-smooth boundary values, the solution of the singularly perturbed subdiffusion equation has a boundarylayer of thickness $\mathcal{O}(ε),$ which brings great challenges to the construction of the efficient numerical schemes. By decomposing the Caputo fractional derivative, the singularly perturbed subdiffusion equation is formally transformed into a class of steady-state diffusive-reaction equation. By means of a kind of tailored finite point method(TFPM) scheme for solving steady-state diffusion-reaction equations and the $\mathcal{L}1$ formula for discretizing the Caputo fractional derivative, we construct a new $\mathcal{L}1$-TFPMscheme for solving singularly perturbed subdiffusion equations. Our proposed numerical scheme satisfies the discrete extremum principle and is unconditionally numerically stable. Besides, we prove that the new TFPM scheme can obtain reliablenumerical solutions as $h ≪ ε$ and $ε ≪ h.$ However, there will be a large error loss dueto the resonance effect as $h ∼ ε.$ Numerical experimental results can demonstrate thevalidity of the numerical scheme.