We consider finite element discretizations for convection–diffusion problems under low regularity assumptions. The derivation and analysis use the primal–dual weak Galerkin (PDWG) finite element framework. The Euler–Lagrange formulation resulting from the PDWG scheme yields a system of equations involving not only the equation for the primal variable but also its adjoint for the dual variable. We show that the proposed PDWG method is stable and convergent. We also derive a priori error estimates for the primal variable in the $H^{ϵ}$-norm for $ϵ\in [0,\frac{1}{2})$. A series of numerical tests that validate the theory is presented as well.

中文翻译：

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对流扩散方程的低正则性原始对偶弱Galerkin有限元方法

在低规则性假设下，我们考虑对流扩散问题的有限元离散化。推导和分析使用原始对偶弱Galerkin（PDWG）有限元框架。PDWG方案产生的Euler-Lagrange公式产生了一个方程系统，不仅涉及原始变量的方程，而且还涉及对偶变量的伴随函数。我们表明，所提出的PDWG方法是稳定且收敛的。我们还导出了原始变量中的先验误差估计$H^{ϵ}$-规范 $ϵ\in [0，\frac{\mathrm{1个}}{\mathrm{2个}}）$。还提出了一系列验证该理论的数值测试。