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Error analysis of variable degree mixed methods for elliptic problems via hybridization


Authors: Bernardo Cockburn and Jayadeep Gopalakrishnan
Journal: Math. Comp. 74 (2005), 1653-1677
MSC (2000): Primary 65N30
DOI: https://doi.org/10.1090/S0025-5718-05-01741-2
Published electronically: March 1, 2005
MathSciNet review: 2164091
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Abstract: A new approach to error analysis of hybridized mixed methods is proposed and applied to study a new hybridized variable degree Raviart-Thomas method for second order elliptic problems. The approach gives error estimates for the Lagrange multipliers without using error estimates for the other variables. Error estimates for the primal and flux variables then follow from those for the Lagrange multipliers. In contrast, traditional error analyses obtain error estimates for the flux and primal variables first and then use it to get error estimates for the Lagrange multipliers. The new approach not only gives new error estimates for the new variable degree Raviart-Thomas method, but also new error estimates for the classical uniform degree method with less stringent regularity requirements than previously known estimates. The error analysis is achieved by using a variational characterization of the Lagrange multipliers wherein the other unknowns do not appear. This approach can be applied to other hybridized mixed methods as well.


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Additional Information

Bernardo Cockburn
Affiliation: School of Mathematics, University of Minnesota, Vincent Hall, Minneapolis, Minnesota 55455
Email: cockburn@math.umn.edu

Jayadeep Gopalakrishnan
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611–8105
Email: jayg@math.ufl.edu

DOI: https://doi.org/10.1090/S0025-5718-05-01741-2
Keywords: Mixed finite elements, hybrid methods, elliptic problems
Received by editor(s): November 26, 2003
Received by editor(s) in revised form: August 1, 2004
Published electronically: March 1, 2005
Additional Notes: The first author is supported in part by the National Science Foundation (Grant DMS-0411254) and by the University of Minnesota Supercomputing Institute
The second author is supported by the National Science Foundation (Grant DMS-0410030).
Article copyright: © Copyright 2005 American Mathematical Society

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