In this paper, we study least-squares finite element methods (LSFEM) for general second-order elliptic equations with nonconforming finite element approximations. The equation may be indefinite. For the two-field potential-flux div LSFEM with Crouzeix-Raviart (CR) element approximation, we present three proofs of the discrete solvability under the condition that mesh size is small enough. One of the proof is based on the coerciveness of the original bilinear form. The other two are based on the minimal assumption of the uniqueness of the solution of the second-order elliptic equation. A counterexample shows that div least-squares functional does not have norm equivalence in the sum space of $H^1$ and CR finite element spaces. Thus it cannot be used as an a posteriori error estimator. Several versions of reliable and efficient error estimators are proposed for the method. We also propose a three-filed potential-flux-intensity div-curl least-squares method with general nonconforming finite element approximations. The norm equivalence in the abstract nonconforming piecewise $H^1$-space is established for the three-filed formulation on the minimal assumption of the uniqueness of the solution of the second-order elliptic equation. The three-filed div-curl nonconforming formulation thus has no restriction on the mesh size, and the least-squares functional can be used as the built-in a posteriori error estimator. Under some restrictive conditions, we also discuss a potential-flux div-curl least-squares method.
翻译:在本文中, 我们研究一般二阶椭圆方程的最小偏差限制元素方法( LSFEM ) 。 方程式可能是无限的。 对于与Crouzeix- Raviart 元素( CR) 元素近似的两地潜在流动的LSFEM div LSFEM 和 Crouzeix- Raviart (CR) 元素近似, 我们提出三种证据, 在网状大小足够小的条件下, 离散的溶解性证明。 其中一种证据是基于原始双线形的强制性。 另两种则基于对二阶椭圆方方程解决方案独特性独特性的最低假设。 对应示例显示, 最小平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面, 平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面,平面平面平面平面平面平面平面平面平面平面平面平面,平面平面平面平面平面平面,平面平面平面平面平面平面,平面平面平面,平面平面平面平面,平面平面平面平面平面,平面,平面平面平面平面平面平面,平面,平面,平面平面平面平面平面平面平面,平面平面平面,平面平面,平面平面平面,平面,平面平面平面平面平面平面平面,平面平面,平面,平面,平面,平面,平面,平面,平面,平面,平面,平面,平面,平面平面,平面平面平面平面平面,平面,平面平面平面平面平面,平面,平面,平面,平面,平面,平面,平面