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This paper is concerned with continuous and discrete approximations of$W^{2,p}$ strong solutions of second-order linear elliptic partial differentialequations (PDEs) in non-divergence form. The continuous approximation of theseequations is achieved through the Vanishing Moment Method (VMM) which adds asmall biharmonic term to the PDE. The structure of the new fourth-order PDE isa natural fit for Galerkin-type methods unlike the original second orderequation since the highest order term is in divergence form. The well-posednessof the weak form of the perturbed fourth order equation is shown as well aserror estimates for approximating the strong solution of the originalsecond-order PDE. A $C^1$ finite element method is then proposed for the fourthorder equation, and its existence and uniqueness of solutions as well asoptimal error estimates in the $H^2$ norm are shown. Lastly, numerical testsare given to show the validity of the method.

Analysis of the Vanishing Moment Method and its Finite Element Approximations for Second-order Linear Elliptic PDEs in Non-divergence Form