Barrier function local and global analysis of an \(L1\) finite element method for a multiterm time-fractional initial-boundary value problem (Q785558)

For the numerical solution of time-fractional initial-boundary value problems, the \(L_1\) discretization for each of the time derivatives on a graded mesh is applied, together with a finite element method (FEM) in space. Using a new discrete stability result, error analysis of this method is performed generalizing the barrier functions and extending it to the multiterm case, which yields sharp bounds for both global and local errors in time. Numerical results for a problem on a one-dimensional spatial domain confirm the theoretical error estimates and the convergence rates for piecewise linear and quadratic finite elements.