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.

中文翻译：

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非发散形式二阶椭圆方程的一种基于最小二乘法的弱伽辽金有限元方法

本文致力于使用离散弱Hessian算子为非发散形式的二阶椭圆方程建立基于最小二乘的弱Galerkin方法。自然，得到的线性系统是对称的和正定的，因此该算法易于实现和分析。H 2 等效范数的收敛分析建立在任意形状的正多边形网格上。当系数矩阵为常数或分段常数时，证明了超收敛结果。数值例子不仅验证了理论结果，而且揭示了一些意想不到的超收敛现象。