Shaohong Du

Shaohong Du, Xiaoping Xie

For adaptive mixed finite element methods (AMFEM), we first introduce the data oscillation to analyze, without the restriction that the inverse of the coefficient matrix of the partial differential equations (PDEs) is a piecewise polynomial matrix, efficiency of the a posteriori error estimator Presented by Carstensen [Math. Comput., 1997, 66: 465-476] for Raviart-Thomas, Brezzi-Douglas-Morini, Brezzi-Douglas-Fortin-Marini elements. Second, we prove that the sum of the stress variable error in a weighted norm and the scaled error estimator is of geometric decay, namely, it reduces with a fixed factor between two successive adaptive loops, up to an oscillation of the right-hand side term of the PDEs. Finally, with the help of this geometric decay, we show that the stress variable error in a weighted norm plus the oscillation of data yields a decay rate in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity

Shaohong Du, Zhiqiang Cai

This paper introduces a new variational formulation for Dirichlet boundary control problem of elliptic partial differential equations, based on observations that the state and adjoint state are related through the control on the boundary of the domain, and that such a relation may be imposed in the variational formulation of the adjoint state. Well-posedness (unique solvability and stability) of the variational problem is established in the $H^{1}(Ω)\times H_{0}^{1}(Ω)$ space for the respective state and adjoint state. A finite element method based on this formulation is analyzed. It is shown that the conforming $k-$th order finite element approximations to the state and the adjoint state, in the respective $L^{2}$ and $H^{1}$ norms converge at the rate of order $k-1/2$ on quasi-uniform mesh for conforming element of order $k$. Numerical examples are presented to validate the theory.

Shaohong Du, Runchang Lin, Zhimin Zhang

We consider mixed finite element approximation of a singularly perturbed fourth-order elliptic problem with two different boundary conditions, and present a new measure of the error, whose components are balanced with respect to the perturbation parameter. Robust residual-based a posteriori estimators for the new measure are obtained, which are achieved via a novel analytical technique based on an approximation result. Numerical examples are presented to validate our theory.

Shaohong Du, Runchang Lin, Zhimin Zhang

In this paper, we investigate adaptive streamline upwind/Petrov Galerkin (SUPG) methods for singularly perturbed convection-diffusion-reaction equations in a new dual norm presented in [Du and Zhang, J. Sci. Comput. (2015)]. The flux is recovered by either local averaging in conforming $H({\rm div})$ spaces or weighted global $L^2$ projection onto conforming $H({\rm div})$ spaces. We further introduce a recovery stabilization procedure, and develop completely robust a posteriori error estimators with respect to the singular perturbation parameter $\varepsilon$. Numerical experiments are reported to support the theoretical results and to show that the estimated errors depend on the degrees of freedom uniformly in $\varepsilon$.