A Least Square Based Weak Galerkin Finite Element Method for Second Order Elliptic Equations in Non-Divergence Form
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
A LEAST SQUARE BASED WEAK GALERKIN FINITE ELEMENT METHOD FOR SECOND ORDER ELLIPTIC EQUATIONS IN NON-DIVERGENCE FORM∗
6+)
†
Peng ZHU (
College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing 314001, China E-mail : [email protected]
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Xiaoshen WANG (
Department of Mathematics and Statistics, University of Arkansas at Little Rock, Little Rock, AR 72204, USA E-mail : [email protected] Abstract This article is devoted to establishing a least square based weak Galerkin method for second order elliptic equations in non-divergence form using a discrete weak Hessian operator. Naturally, the resulting linear system is symmetric and positive definite, and thus the algorithm is easy to implement and analyze. Convergence analysis in the H 2 equivalent norm is established on an arbitrary shape regular polygonal mesh. A superconvergence result is proved when the coefficient matrix is constant or piecewise constant. Numerical examples are performed which not only verify the theoretical results but also reveal some unexpected superconvergence phenomena. Key words
least square based weak Galerkin method; non-divergence form; weak Hessian operator; polygonal mesh
2010 MR Subject Classification
1
65N15; 65N30
Introduction
Let Ω be a bounded convex domain in Rd (d = 2, 3) with a Lipschitz continuous boundary ∂Ω. In this paper, we will consider the following second order elliptic problems in non-divergence form: A : D2 u = f in Ω, (1.1) u = 0 on ∂Ω, ∗ Received
April 6, 2018; revised February 2, 2020. The first author was supported by Zhejiang Provincial Natural Science Foundation of China (LY19A010008). † Corresponding author: Peng ZHU.
1554
ACTA MATHEMATICA SCIENTIA
Vol.40 Ser.B
where the coefficient tensor A(x) = {aij (x)}d×d is assumed to be symmetric, uniformly bounded 2 and positive definite. Here D2 u = {∂ij u}d×d is the Hessian matrix of u, and A : D2 u = d P 2 aij ∂ij u. i,j=1
Elliptic problems in non-divergence form have applications in stochastic processes and game theory. The problems can not be rewritten in divergence form when coefficients aij are nonsmooth. The non-divergence form of (1.1) makes it almost impossible to have a weak formulation, and thus it is difficult to derive and analyze the finite element methods for solving this PDE. To overcome this difficulty, quite a few papers have been devoted in recent years to finite element methods for solving this equation using various special kinds of treatments (e.g. [1– 3, 6, 8, 11]). The least square method is a general method which can be used to find the best approximation of a given function from a vector space with respect to a certain inner product. Thus, the resulting linear system is always symmetric and positive definite. Among the references mentioned above, [8] has the flavor of the least square method and [6, 8] are least square based methods. A least square formulation of this problem can be describe
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