Superconvergence and $L^{\infty}$-Error Estimates of RT1 Mixed Methods for Semilinear Elliptic Control Problems with an Integral Constraint
Year: 2012
Numerical Mathematics: Theory, Methods and Applications, Vol. 5 (2012), Iss. 3 : pp. 423–446
Abstract
In this paper, we investigate the superconvergence property and the $L^{\infty}$-error estimates of mixed finite element methods for a semilinear elliptic control problem with an integral constraint. The state and co-state are approximated by the order one Raviart-Thomas mixed finite element space and the control variable is approximated by piecewise constant functions or piecewise linear functions. We derive some superconvergence results for the control variable and the state variables when the control is approximated by piecewise constant functions. Moreover, we derive $L^{\infty}$-error estimates for both the control variable and the state variables when the control is discretized by piecewise linear functions. Finally, some numerical examples are given to demonstrate the theoretical results.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/nmtma.2012.m1118
Numerical Mathematics: Theory, Methods and Applications, Vol. 5 (2012), Iss. 3 : pp. 423–446
Published online: 2012-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 24
Keywords: Semilinear elliptic equations optimal control problems superconvergence $L^{\infty}$-error estimates mixed finite element methods postprocessing.