Abstract In this paper we formulate and analyze adaptive (space-time) least-squares finite element methods for the solution of convection-diffusion equations. The convective derivative π β β β‘ u v u is considered as part of the total time derivative d d β’ t β’ u = β t β‘ u + π β β β‘ u d{dtu=βu+v u}, and therefore we can use a rather standard stability and error analysis for related space-time finite element methods. For stationary problems we restrict the ansatz space H 0 1 β’ (Ξ©) H^{1β () } such that the convective derivative is considered as an element of the dual H - 1 β’ (Ξ©) H^{-1 () } of the test space H 0 1 β’ (Ξ©) H^{1β () }, which also allows unbounded velocities π v. While the discrete finite element schemes are always unique solvable, the numerical solutions may suffer from a bad approximation property of the finite element space when considering convection dominated problems, i. e. , small diffusion coefficients. Instead of adding suitable stabilization terms, we aim to resolve the solutions by using adaptive (space-time) finite element methods. For this we introduce a least-squares approach where the discrete adjoint defines local a posteriori error indicators to drive an adaptive scheme. Numerical examples illustrate the theoretical considerations.
KΓΆthe et al. (Fri,) studied this question.