Methods and Applications of Analysis

Volume 26 (2019)

Number 2

Special Issue in Honor of Roland Glowinski (Part 1 of 2)

Guest Editors: Xiaoping Wang (Hong Kong University of Science and Technology) and Xiaoming Yuan (The University of Hong Kong)

Analysis of the vanishing moment method and its finite element approximations for second-order linear elliptic PDEs in non-divergence form

Pages: 167 – 194

DOI: https://dx.doi.org/10.4310/MAA.2019.v26.n2.a5

Authors

Xiaobing Feng (Department of Mathematics, University of Tennessee, Knoxville, Tn., U.S.A.)

Thomas Lewis (Department of Mathematics and Statistics, University of North Carolina, Greensboro, N.C., U.S.A.)

Stefan Schnake (Department of Mathematics, University of Oklahoma, Norman, Ok., U.S.A.)

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.

Keywords

elliptic PDEs in non-divergence form, strong solution, vanishing moment method, $C^1$ finite element method, discrete Calderon–Zygmund estimate

2010 Mathematics Subject Classification

65N12, 65N15, 65N30

Received 28 December 2017

Accepted 12 April 2019

Published 2 April 2020