Unconditionally optimal H-error estimate of a fast nonuniform L2-1 scheme for nonlinear subdiffusion equations.

This paper is concerned with the unconditionally optimal H-1-error estimate of a fast second-order scheme for solving nonlinear subdiffusion equations on the nonuniform mesh. We use the Galerkin finite element method (FEM) to discretize the spacial direction, the Newton linearization method to approximate the nonlinear term and the sum-of-exponentials (SOE) approximation to speed up the evaluation of Caputo derivative. Our analysis of the unconditionally optimal H-1-error estimate involves the temporal-spatial error splitting approach, an improved discrete fractional Gronwall inequality and error convolution structure. In order to find a suitable test function to estimate H-1-error, we here consider two cases: linear and high-order ELM space, using the time-discrete operator and Laplace operator in the test function respectively. Numerical tests are provided demonstrate the effectiveness and the unconditionally optimal H-1-error convergence of our scheme.