This article presents a new primal-dual weak Galerkin method for second order elliptic equations in non-divergence form. The new method is devised as a constrained $L^p$-optimization problem with constraints that mimic the second order elliptic equation by using the discrete weak Hessian locally on each element. An equivalent min-max characterization is derived to show the existence and uniqueness of the numerical solution. Optimal order error estimates are established for the numerical solution under the discrete $W^{2,p}$ norm, as well as the standard $W^{1,p}$ and $L^p$ norms. An equivalent characterization of the optimization problem in term of a system of fixed-point equations via the proximity operator is presented. An iterative algorithm is designed based on the fixed-point equations to solve the optimization problems. Implementation of the iterative algorithm is studied and convergence of the iterative algorithm is established. Numerical experiments for both smooth and non-smooth coefficients problems are presented to verify the theoretical findings.
翻译:本条以非调和形式为第二顺序椭圆方程式提出了一种新的原始弱加勒金方法。新方法设计为限制的美元=p$-优化问题,其限制因素通过在每一元素上使用离散的弱黑森对等方程式,模仿第二顺序椭圆方程式。提出了相当的微积分特性,以显示数字解决办法的存在和独特性。在离散的 $+%2,p}规范下,为数字解决方案确定了最优顺序错误估计,以及标准 $W$1,p}$和$L ⁇ p$规范。介绍了通过近距离操作器对固定点方程式系统术语的最优化问题的同等定性。根据固定点方程式设计的迭代算算法,以解决优化问题。对迭代算法的应用进行了研究,并确定了迭代算法的趋同性。对光度和非毛系数问题进行了数值实验,以核实理论结论。