A Cost-Efficient Space-Time Adaptive Algorithm for Coupled Flow and Transport

Galerkin Orthogonality for Temporal and Spatial Error of Transport Problem

For the temporal error e = u − u τ , we get the following Galerkin orthogonality by subtracting equation (2.7) from equation (2.3):

with a non-vanishing right-hand side term depending on the temporal error in the approximation of the flow field. For the spatial error e = u τ − u τ ⁢ h , we get the following Galerkin orthogonality by subtracting equation (2.10) from equation (2.7):

with a non-vanishing right-hand side term depending on the stabilization and the spatial error in the approximation of the flow field.

Transport: Dual Problems and Residuals

Within the context of the DWR philosophy, the dual problems are generally given as optimality or stationary conditions regarding the underlying Lagrangian functionals. More precisely, considering the directional derivatives of the Lagrangian functionals (3.3), also known as Gâteaux derivatives (cf., e.g., [11]), with respect to their first argument, i.e.,

leads to the following dual transport problems: find the continuous dual solution z ∈ X , the semi-discrete dual solution z τ ∈ X τ r and the fully discrete dual solution z τ ⁢ h ∈ X τ ⁢ h r , p , respectively, such that

where we refer to our works [9, 14] for a detailed description of the adjoint bilinear forms A ′ , A τ ′ , A S ′ as well as the dual right-hand side term J ′ .

The primal and dual residuals based on the continuous and semi-discrete-in-time schemes are defined by means of the Gâteaux derivatives of the Lagrangian functionals in the following way:

Flow: Dual Problems and Residuals

For the sake of completeness, considering the directional derivatives of the Lagrangian functionals (3.1), also known as Gâteaux derivatives (cf., e.g., [11]), with respect to their first argument, i.e.,

leads to the following dual flow problems, although these problems were not used within the underlying cost reduced approach here: find the continuous dual flow solution z = { w , q } ∈ Y , the semi-discrete dual flow solution z σ = { w σ , q σ } ∈ Y σ r and the fully discrete dual flow solution z σ ⁢ h ⁢ { w σ ⁢ h , q σ ⁢ h } ∈ Y σ ⁢ h r , p , respectively, such that

where we refer to [14] for a detailed description of the adjoint bilinear forms B ′ , B σ ′ , B σ ⁢ h ′ as well as the dual right-hand side term J ′ .

Finally, the primal and dual residuals based on the semi-discrete-in-time schemes are defined by means of the Gâteaux derivatives of the Lagrangian functionals in the following way: