Adaptive Least-Squares (Space-Time) Finite Element Methods For Convection-Diffusion Problems

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 𝒗 ⋅ ∇ ⁡ u v u is considered as part of the total time derivative d d ⁢ t ⁢ u = ∂ t ⁡ u + 𝒗 ⋅ ∇ ⁡ u d{dtu=ₓu+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 ⁢ (Ω) H^{1₀ () } such that the convective derivative is considered as an element of the dual H - 1 ⁢ (Ω) H^{-1 () } of the test space H 0 1 ⁢ (Ω) H^{1₀ () }, which also allows unbounded velocities 𝒗 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.