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One of the main tools in the proof of residual-based a posteriori error
estimates is a quasi-interpolation operator due to Clément.
We modify this operator in the setting of a partition of unity
with the effect that the approximation error has a local average zero.
This results in a new residual-based a posteriori error estimate
with a volume contribution which is smaller than in the standard estimate.
For an elliptic model problem, we discuss applications to conforming,
nonconforming and mixed...
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