A conforming discontinuous Galerkin (CDG) finite element method is designed for solving second order non-self adjoint and indefinite elliptic equations. Unlike other discontinuous Galerkin (DG) methods, the numerical trace on the edge/triangle between two elements is not the average of two discontinuous \(P_k\) functions, but a lifted \(P_{k+2}\) function from four (eight in 3D) nearby \(P_k\) functions. While all existing DG methods have the optimal order of convergence, this CDG method has a superconvergence of order two above the optimal order when solving general second order elliptic equations. Due to the superconvergence, a post-process lifts a \(P_k\) CDG solution to a quasi-optimal \(P_{k+2}\) solution on each element. Numerical tests in 2D and 3D are provided confirming the theory.

设计了一种用于求解二阶非自伴随和不定椭圆方程的一致间断伽辽金（CDG）有限元方法。与其他不连续伽辽金 (DG) 方法不同，两个元素之间的边/三角形上的数值轨迹不是两个不连续\(P_k\)函数的平均值，而是从四个不连续\(P_{k+2}\) 函数提升的 \(P_{k+2}\)函数（3D 中的八个）附近的\(P_k\)个函数。虽然所有现有的 DG 方法都具有最佳收敛阶数，但该 CDG 方法在求解一般二阶椭圆方程时具有高于最佳阶数的二阶超收敛性。由于超收敛，后处理将\(P_k\) CDG 解提升为每个元素上的准最优\(P_{k+2}\)解。提供了 2D 和 3D 数值测试来证实该理论。