A linear Galerkin numerical method for a quasilinear subdiffusion equation

Error of the Galerkin scheme for a semilinear subdiffusion equation with time-dependent coefficients and nonsmooth data

“…Combining the above with the error decomposition (11) we obtain the final error estimate for the fully discrete scheme.…”

“…Now choosing v h = e n h , utilizing the (δ α t y n , y n ) ≥ 1 2 δ α t y n 2 inequality for the L 2 functions (see Proposition 2 in [11]), noticing that a is a positive-definite form, and using the Cauchy-Schwarz inequality, we get…”

“…This approximation for the Caputo derivative is based on a piecewise linear basis and was introduced in [10] to compute the Riemann-Liouville derivative. Further analyses of this method can be found, for example, in [11] where the optimal error constant for smooth functions was found. For such a regularity, the L1 method achieves a global in time order 2 − α that can deteriorate to mere α for Hölder functions.…”

“…A completely different approach was taken in [18] where an energy-type approach was developed to tackle the time-dependent diffusivity case which causes severe difficulties. This approach was later refined in a paper by the first author, [9] in which all assumptions were substantially weakened and the result was generalized to the semilinear case, building a passage to the quasilinear problem (which was analyzed in [11] in the case of regular initial data). However, in all of these papers, the devised numerical methods were semi-discrete.…”

“…where u h (x, 0) = P h ϕ(x) is the orthogonally projected initial condition into the space V h ⊂ H 1 0 of piecewise linear continous functions over Ω h ⊂ Ω, that is, the quasiuniform triangulation of Ω with maximal diameter h. Of course, dim V h < ∞. Next, introduce the time grid t n = n∆t with ∆t = T /N and use the L1 scheme for the Caputo derivative denoted δ α t defined as [11]…”