This work was partly supported by the NSF under grants DMS-0411254 and DMS-0410030
Mathematics of Computation
Finite element method, Error analysis (Mathematics), Numerical analysis, Differential equations, Elliptic -- Numerical solutions, Multipliers (Mathematical analysis)
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.
Cockburn, B. and Gopalakrishnan, J. (2005). Error Analysis of Variable Degree Mixed Methods for Elliptic Problems via Hybridization. Mathematics of Computation, Vol. 74, No. 252, pp. 1653-1677.