论文标题
一般二阶椭圆方程的最小二乘方法具有不合格的有限元
Least-Squares Methods with Nonconforming Finite Elements for General Second-Order Elliptic Equations
论文作者
论文摘要
在本文中,我们研究了最小二乘有限元方法(LSFEM),用于具有不合格有限元近似值的一般二阶椭圆方程。方程可能是不确定的。对于具有Crouzeix-raviart(CR)元素近似的两场电势频率Div LSFEM,我们在网格大小足够小的情况下提供了三个证明离散可溶性的证明。证明之一是基于原始双线性形式的强制性。另外两个是基于对二阶椭圆方程溶液独特性的最小假设。反例显示,Div最小二乘功能在$ H^1 $的总和和CR有限元空间中没有标准等效性。因此,它不能用作后验误差估计器。为该方法提出了几种可靠且有效的误差估计器的版本。我们还提出了一种具有一般不合格的有限元近似值的三个文件势液强度最小二乘法。对于三阶椭圆方程解决方案的唯一性,为三个文件的公式建立了抽象不合格的分段$ h^1 $空间中的规范等效性。因此,三个文件的Div-Curl不合格公式对网格大小没有限制,并且最小二乘功能可以用作内置的后验误差估计器。在某些限制性条件下,我们还讨论了潜在的Flux Div-Curl最小二乘法。
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.