A primal-dual weak Galerkin finite element method for second order elliptic equations in non-divergence form
A primal-dual weak Galerkin finite element method for second order elliptic equations in non-divergence form
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DOI：
10.1090/mcom/3220
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发表时间：
2015-10
期刊：
影响因子：
--
通讯作者：
Chunmei Wang;Junping Wang
中科院分区：
文献类型：
--
作者：
Chunmei Wang;Junping Wang
本文提出了一种新的非散度型二阶椭圆方程的数值求解算法。该方法基于一个离散的弱Hessian算子，通过遵循弱Galerkin策略局部构造。数值解的特征是利用离散弱Hessian模拟二阶椭圆方程的约束的非负二次泛函的最小化。由此得到的欧拉-拉格朗日方程提供了一种对称的有限元方案，其中包括被称为拉格朗日乘子的原始变量和对偶变量，因此被称为原始-对偶弱伽辽金有限元方法。对于离散$H^2$范数的有限元近似，以及通常的$H^1$和$L^2$范数，导出了最优阶误差估计。给出了在凸域和非凸域上光滑系数和非光滑系数的一些数值结果。
This article proposes a new numerical algorithm for second order elliptic equations in non-divergence form. The new method is based on a discrete weak Hessian operator locally constructed by following the weak Galerkin strategy. The numerical solution is characterized as a minimization of a non-negative quadratic functional with constraints that mimic the second order elliptic equation by using the discrete weak Hessian. The resulting Euler-Lagrange equation offers a symmetric finite element scheme involving both the primal and a dual variable known as the Lagrange multiplier, and thus the name of primal-dual weak Galerkin finite element method. Optimal order error estimates are derived for the finite element approximations in a discrete $H^2$-norm, as well as the usual $H^1$- and $L^2$-norms. Some numerical results are presented for smooth and non-smooth coefficients on convex and non-convex domains.