Dual least-squares finite element method with stabilization

The LL*-method is a least-squares finite element approach producing an approximation by solving dual problem corresponding to the given partial differential equations. Due to the unique structure of LL* approximation, it has advantages if the problem has low regularities and when L²-approximation needs to be established. As a drawback, piecewise polynomial type approximation often generates artifacts such as spurious oscillations near where shocks or discontinuities occur in solution. Allowing discontinuous piecewise polynomial approximation in LL* seems to exacerbate this trouble. This paper presents a stabilized LL*-method that is designed to effectively reduce these oscillatory behavior. The consistency and error convergence of proposed method are analyzed and numerical examinations are performed.