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Title: An a priori error analysis of adjoint-based super-convergent Galerkin approximations of linear functionals
Abstract We present the first a priori error analysis of a new method proposed in Cockburn & Wang (2017, Adjoint-based, superconvergent Galerkin approximations of linear functionals. J. Comput. Sci., 73, 644–666), for computing adjoint-based, super-convergent Galerkin approximations of linear functionals. If $J(u)$ is a smooth linear functional, where $$u$$ is the solution of a steady-state diffusion problem, the standard approximation $$J(u_h)$$ converges with order $$h^{2k+1}$$, where $$u_h$$ is the Hybridizable Discontinuous Galerkin approximation to $$u$$ with polynomials of degree $k>0$. In contrast, numerical experiments show that the new method provides an approximation that converges with order $$h^{4k}$$, and can be computed by only using twice the computational effort needed to compute $$J(u_h)$$. Here, we put these experimental results in firm mathematical ground. We also display numerical experiments devised to explore the convergence properties of the method in cases not covered by the theory, in particular, when the solution $$u$$ or the functional $$J(\cdot )$$ are not very smooth. We end by indicating how to extend these results to the case of general Galerkin methods.  more » « less
Award ID(s):
1912646
PAR ID:
10402612
Author(s) / Creator(s):
;
Date Published:
Journal Name:
IMA Journal of Numerical Analysis
Volume:
42
Issue:
2
ISSN:
0272-4979
Page Range / eLocation ID:
1050 to 1086
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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