A convincing feature of least-squares finite element methods is the built-in a posteriori error estimator for any conforming discretization. In order to generalize this property to discontinuous finite element ansatz functions, this paper introduces a least-squares principle on piecewise Sobolev functions for the solution of the Poisson model problem in 2D with mixed boundary conditions. It allows for fairly general discretizations including standard piecewise polynomial ansatz spaces on triangular and polygonal meshes. The presented scheme enforces the interelement continuity of the piecewise polynomials by additional least-squares residuals. A side condition on the normal jumps of the flux variable requires a vanishing integral mean and enables a natural weighting of the jump in the least-squares functional in terms of the mesh size. This avoids over-penalization with additional regularity assumptions on the exact solution as usually present in the literature on discontinuous LSFEM. The proof of the built-in a posteriori error estimation for the over-penalized scheme is presented as well. All results in this paper are robust with respect to the size of the domain guaranteed by a suitable weighting of the residuals in the least-squares functional. Numerical experiments exhibit optimal convergence rates of the adaptive mesh-refining algorithm for various polynomial degrees.

翻译：最小二乘有限元方法的一个令人信服的特征是，对于任意协调离散化，其具有内嵌的后验误差估计器。为了将该性质推广至不连续有限元试探函数，本文针对二维混合边界条件下的泊松模型问题，在分片Sobolev函数空间上引入了一个最小二乘原理。该方法允许相当一般的离散化，包括三角形和多边形网格上的标准分片多项式试探空间。所提出的方案通过额外的最小二乘残差强制分片多项式的单元间连续性。通量变量法向跳跃的一个附加条件要求其积分均值为零，这使得最小二乘泛函中的跳跃项能够根据网格尺寸进行自然加权。这避免了像不连续LSFEM文献中通常存在的，通过附加精确解的正则性假设进行过度惩罚。本文还给出了过度惩罚方案的内嵌后验误差估计的证明。通过最小二乘泛函中残差的适当加权，本文的所有结果在域尺寸方面具有鲁棒性。数值实验表明，对于不同多项式次数，自适应网格加密算法达到了最优收敛率。