Finite element algorithms for transport-diffusion problems: stability, adaptivity, tractability

Abstract

Partial differential equations with nonnegative characteristic form arise in numerous mathematical models of physical phenomena: stochastic analysis, in particular, is a fertile source of equations of this kind. We survey recent developments concerning the finite element approximation of these equations, focusing on three relevant aspects: (a) stability and stabilisation; (b) hp-adaptive algorithms driven by residual-based a posteriori error bounds, capable of automatic variation of the granularity h of the finite element partition and of the local polynomial degree p; (c) complexity-reduction for high-dimensional transport-diffusion problems by stabilised sparse finite element methods.