A Posteriori Error Analysis for Variable-Coefficient Multiterm Time-Fractional Subdiffusion Equations

Abstract

An initial-boundary value problem of subdiffusion type is considered; the temporal component of the differential operator has the form \(\sum _{i=1}^{\ell }q_i(t)\, D _t ^{\alpha _i} u(x,t)\), where the \(q_i\) are continuous functions, each \(D _t ^{\alpha _i}\) is a Caputo derivative, and the \(\alpha _i\) lie in (0, 1]. Maximum/comparison principles for this problem are proved under weak hypotheses. A new positivity result for the multinomial Mittag-Leffler function is derived. A posteriori error bounds are obtained in \(L_2(\Omega )\) and \(L_\infty (\Omega )\), where the spatial domain \(\Omega \) lies in \({\mathbb {R}}^d\) with \(d\in \{1,2,3\}\). An adaptive algorithm based on this theory is tested extensively and shown to yield accurate numerical solutions on the meshes generated by the algorithm.