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 = 1 ℓ q i ( t ) D t α i u ( x , t ) , where the q i are continuous functions, each D t α i 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 L 2 ( 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.