A new primal-dual weak Galerkin (PDWG) finite element method is introduced and analyzed for the ill-posed elliptic Cauchy problems with ultra-low regularity assumptions on the exact solution. The Euler-Lagrange formulation resulting from the PDWG scheme yields a system of equations involving both the primal equation and the adjoint (dual) equation. The optimal order error estimate for the primal variable in a low regularity assumption is established. A series of numerical experiments are illustrated to validate effectiveness of the developed theory.

翻译：引入并分析一种新的原始二元弱Galerkin(PDWG)有限元素法(PDWG),该元素法针对对精确解决方案假设的超低常规性假设存在不良的椭圆形粘结问题。PDWG方案产生的Euler-Lagrange配方产生一个包含原始方程和联合(双)方程的方程系统。在低常规假设中确定了原始变量的最佳顺序误差估计。一系列数字实验被演示为证实发达理论的有效性。