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In this paper, a stabilized finite element method for optimal control problems governed by a convection dominated diffusion equation is investigated. The state and the adjoint variables are approximated by piecewise linear continuous functions with bubble functions. The control variable either is approximated by piecewise linear functions (called the standard method) or is not discretized directly (called the variational discretization method). The stabilization term only depends on bubble functions, and the projection operator can be replaced by the difference of two local Gauss integrations. A priori error estimates for both methods are given and numerical examples are presented to illustrate the theoretical results. [ABSTRACT FROM PUBLISHER]

We consider a boundary value problem involving a Riemann–Liouville fractional derivative of order α ∈ ( 3 / 2 , 2 ) on the unit interval ( 0 , 1 ) . The standard Galerkin finite element approximation converges slowly due to the presence of singularity term x α − 1 in the solution representation. In this work, we develop a simple technique, by transforming it into a second-order two-point boundary value problem with nonlocal low order terms, whose solution can reconstruct directly the solution to the original problem. The stability of the variational formulation, and the optimal regularity pickup of the solution are analyzed. A novel Galerkin finite element method with piecewise linear or quadratic finite elements is developed, and L 2 ( D ) error estimates are provided. The approach is then applied to the corresponding fractional Sturm–Liouville problem, and error estimates of the eigenvalue approximations are given. Extensive numerical results fully confirm our theoretical study. [ABSTRACT FROM AUTHOR]

In this paper, an inverse Robin problem for an elliptic equation is investigated. The Robin coefficient is assumed to be a small perturbation from a reference coefficient, hence a sparse structure is a priori known. An Lq-regularized output least square formulation is considered with q ∈ (0,1). The existence of the minimizer is proved. The convergence of the finite element discretization with respect to mesh size is shown. A modified Newton algorithm is applied to solve the smoothing first order optimality system. Two-dimensional and three-dimensional numerical examples demonstrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this article, theP2−P2-stabilized finite element method based on two local Gaussian quadratures is applied to discretize the Stokes eigenvalue problem, and the corresponding convergence analysis is given. Furthermore, a two-level scheme, which solves the Stokes eigenvalue problem on a coarser grid and a Stokes problem on the fine grid, is employed to reduce the computational cost. Numerical examples are given to confirm the theoretical results. [ABSTRACT FROM PUBLISHER]

We present a finite element analysis of electrical impedance tomography for reconstructing the conductivity distribution from electrode voltage measurements by means of Tikhonov regularization. Two popular choices of the penalty term, i.e., the H1(Ω)-norm smoothness penalty and total variation seminorm penalty, are considered. A piecewise linear finite element method is employed for discretizing the forward model, i.e., the complete electrode model, the conductivity, and the penalty functional. The convergence of the finite element approximations for the Tikhonov model on both polyhedral and smooth curved domains is established. This provides rigorous justifications for the ad hoc discretization procedures. Numerical experiments confirm the convergence analysis. [ABSTRACT FROM AUTHOR]