ANALYSIS OF THE VANISHING MOMENT METHOD AND ITS FINITE ELEMENT APPROXIMATIONS FOR SECOND-ORDER LINEAR ELLIPTIC PDES IN NON-DIVERGENCE FORM

This paper is concerned with continuous and discrete approximations of W-2,W-p strong solutions of second-order linear elliptic partial differential equations (PDEs) in non-divergence form. The continuous approximation of these equations is achieved through the Vanishing Moment Method (VMM) which adds a small biharmonic term to the PDE. The structure of the new fourth-order PDE is a natural fit for Galerkin-type methods unlike the original second order equation since the highest order term is in divergence form. The well-posedness of the weak form of the perturbed fourth order equation is shown as well as error estimates for approximating the strong solution of the original second-order PDE. A C-1 finite element method is then proposed for the fourth order equation, and its existence and uniqueness of solutions as well as optimal error estimates in the H-2 norm are shown. Lastly, numerical tests are given to show the validity of the method.