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end{cases} (21) No restriction on Δt has to be imposed for each evaporation sub-step because the temporal integration on the evaporation term is carried out with the analytical solutions.

For quasi-optimal rates of the discretization error in the sense of   Theorem  4.5, additional regularity of the data has to be imposed for higher-order elements p ≥ 1, cf.   Consequence  5.2.

Two major slides have occurred since 1970, the first in 1974 and the second in 1986; in both cases one-way traffic had to be imposed for a time in the affected area.

SYRIZA and the European Left argue that in the framework of a European agreement, the largest part of the public debt's nominal value has to be written off, a moratorium on its repayment has to be imposed, and a growth clause for servicing the remaining debt has to be introduced, in order to use spare resources towards growth.

Theorem  4.5   proves that for lowest-order elements p = 1, even optimal rates for the discretization error are achieved, while for higher-order elements p ≥ 2 additional regularity of f has to be imposed, e.g., f ∈ H 1 for p = 2.  □ Numerical examples for the 2D Laplacian with mixed Dirichlet Neumann boundary conditions are found in  [82] together with a detailed discussion of the implementation.

Theorem  4.5   proves that for lowest-order elements p = 1, even optimal rates for the discretization error are achieved, while for higher-order elements p ≥ 2 additional regularity of f has to be imposed, e.g., f ∈ H 1 for p = 2.  □ The adaptive algorithm leads to convergence with quasi-optimal rate for the estimator η in the sense of   Theorem  4.1.

For lowest-order elements p = 1,   Theorem  8.5   states optimal rates for the discretization error, while for higher-order elements p ≥ 1, additional regularity of f has to be imposed, e.g., f ∈ H 1 for p = 2.  □ The adaptive algorithm leads to convergence with quasi-optimal rate for the estimator ϱ in the sense of   Theorem  8.4.

For lowest-order elements p = 1,   Theorem  8.5   states optimal rates for the discretization error, while for higher-order elements p ≥ 1, additional regularity of f has to be imposed, e.g., f ∈ H 1 for p = 2.  □ In this section, we give three examples of adaptive FEM for nonlinear problems.

As we shall see, more constraints may have to be imposed, if for instance the flow in the pipe is required to be laminar.

The Department of Transportation has suggested that "congestion fees" be imposed for landings, which would vary based on traffic and time of day.

And the United Steelworkers' union has asked for duties to be imposed on Chinese tyre imports under a little-used law known as Section 421.