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Published in:
01-07-2020
Barrier Function Local and Global Analysis of an L1 Finite Element Method for a Multiterm Time-Fractional Initial-Boundary Value Problem
Published in: Journal of Scientific Computing | Issue 1/2020
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Abstract
An initial-boundary value problem of the form \(\sum _{i=1}^{l}q_i D _t ^{\alpha _i} u(x,t)- \Delta u =f\) is considered, where each \(D _t ^{\alpha _i}\) is a Caputo fractional derivative of order \(\alpha _i\in (0,1)\) and the spatial domain \(\Omega \) lies in \(\mathbb {R}^d\) for \(d\in \{1,2,3\}\). To solve the problem numerically, we apply the L1 discretisation to each fractional derivative on a graded temporal mesh, together with a standard finite element method for the spatial derivatives on a quasiuniform spatial mesh. A new proof of the stability of this method, which is more complicated than the \(l=1\) case of a single fractional derivative, is given using barrier functions; this powerful new technique leads to sharp error estimates in \(L^2(\Omega )\) and \(H^1(\Omega )\) at each time level \(t_m\) that show precisely the improvement in accuracy of the method as one moves away from the initial time \(t=0\). Consequently, while for global optimal accuracy one needs a mesh that is strongly graded when all the \(\alpha _i\) are near zero, for local optimal accuracy away from \(t=0\) one needs a much less severe mesh grading. Numerical experiments show the sharpness of our theoretical results.