A Preconditioning Technique for an All-at-once System from Volterra Subdiffusion Equations with Graded Time Steps

Volterra subdiffusion problems with weakly singular kernel describe the dynamics of processes well. The graded L1 scheme is often chosen to discretize such since it can handle singularity solution near $$t = 0$$ . In this paper, we propose a modification. We first split time interval [0, T] into $$[0, T_0]$$ and $$[T_0, T]$$ , where $$T_0$$ ( $$0< T_0 < T$$ ) reasonably small. Then, applied in while uniform one used Our all-at-once system derived based on strategy. order solve arising efficiently, two subproblems design preconditioners. Some properties these preconditioners are also investigated. Moreover, extend our method semilinear problems. Numerical results reported show efficiency method.