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

TL;DR

This paper investigates the Vanishing Moment Method for second-order elliptic PDEs in non-divergence form, proposing a finite element approach with error analysis and numerical validation.

Contribution

It introduces a novel finite element method for the Vanishing Moment approximation of non-divergence form PDEs, with proven error estimates and numerical results.

Findings

The method achieves optimal error estimates in the H^2 norm.

Numerical tests confirm the effectiveness of the proposed finite element approach.

The Vanishing Moment Method provides a stable and accurate approximation for second-order elliptic PDEs.

Abstract

This paper is concerned with continuous and discrete approximations of $W^{2,p}$ 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.