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

An initial-boundary value problem of subdiffusion type is considered; the temporal component of the differential operator has the form ∑i=1ℓqi(t)Dtαiu(x,t), where the q_i are continuous functions, each Dtαi is a Caputo derivative, and the α_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 L₂(Ω) and L_∞(Ω) , where the spatial domain Ω lies in R^d with d∈ { 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.