Quasi-Interpolation and A Posteriori Error Analysis in Finite Element Methods

• Published in: ESAIM: Mathematical Modelling and Numerical Analysis Vol. 33; no. 6; pp. 1187 - 1202
• Main Author: Carstensen, Carsten
• Format: Journal Article
• Language: English
• Published: Les Ulis EDP Sciences 01.11.1999
• Subjects: 65N30; 65R20; 73C50; adaptive algorithm; A posteriori error estimates; Classical and quantum physics: mechanics and fields; Classical mechanics of continuous media: general mathematical aspects; Elasticity, plasticity: general mathematical aspects; Error analysis; Exact sciences and technology; Integral equations, integral transforms; Mathematics; mixed finite element method; nonconforming finite element method; Numerical analysis; Numerical analysis. Scientific computation; Partial differential equations, boundary value problems; Physics; reliability; Sciences and techniques of general use; Integral operator; Stability; Adaptive algorithm; Error estimation; Lipschitz partition of unity; Mixed method; Nonconformal method; Lipschitz continuity; Finite element method; Interpolation; A posteriori estimation; Dirichlet problem; Reliability; Conformal method
• ISSN: 0764-583X; 1290-3841
• DOI: 10.1051/m2an:1999140

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 finite element methods.