\( \alpha \)-robust error analysis of a mixed finite element method for a time-fractional biharmonic equation (Q2041531)

The authors consider the numerical resolution of the linear time fractional biharmonic equation \({D}_t^{\alpha } u+{\Delta }^2u-c{\Delta } u =f\). The fractional derivative is given in the sense of Caputo and its order is satisfying \(\alpha \in (0,1)\). The boundary conditions are \(u = \Delta u = 0\) on \(\partial \Omega \). Some a priori estimates on the exact solution are derived. To perform a numerical scheme, the problem is written under the form of a system of two problems using the new variable \(p=-\Delta u\). A weak formulation is then derived using this system in which the space of the test functions and the solutions is \(H^1_0\). Then, a finite element method is applied to the system. The discretization in time is performed using the well-known L1 scheme on a graded meshes. To analyze the numerical obtained scheme, a convenient discrete Gronwall inequality is derived and proved. A discrete stability is proved and an error estimate is shown as well. Some numerical tests are presented to support the theoretical results. The article is interesting and it merits to read.