This paper studies adaptive first-order least-squares finite element methods for second-order elliptic partial differential equations in non-divergence form. Unlike the classical finite element method which uses weak formulations of PDEs not applicable for the non-divergence equation, the first-order least-squares formulations naturally have stable weak forms without using integration by parts, allow simple finite element approximation spaces, and have build-in a posteriori error estimators for adaptive mesh refinements. The non-divergence equation is first written as a system of first-order equations by introducing the gradient as a new variable. Then two versions of least-squares finite element methods using simple $C^0$ finite elements are developed in the paper, one is the $L^2$-LSFEM which uses linear elements, the other is the weighted-LSFEM with a mesh-dependent weight to ensure the optimal convergence. Under a very mild assumption that the PDE has a unique solution, optimal a priori and a posteriori error estimates are proved. With an extra assumption on the operator regularity which is weaker than traditionally assumed, convergences in standard norms for the weighted-LSFEM are also discussed. $L^2$-error estimates are derived for both formulations. We perform extensive numerical experiments for smooth, non-smooth, and even degenerate coefficients on smooth and singular solutions to test the accuracy and efficiency of the proposed methods.

中文翻译：

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非发散形式二阶椭圆方程的自适应一阶系统最小二乘有限元方法

本文研究了非发散形式的二阶椭圆偏微分方程的自适应一阶最小二乘有限元方法。与使用不适用于非发散方程的偏微分方程弱公式的经典有限元方法不同，一阶最小二乘公式自然具有稳定的弱形式，无需使用分部积分，允许简单的有限元近似空间，并具有构建- 在用于自适应网格细化的后验误差估计中。通过引入梯度作为新变量，非发散方程首先被写成一阶方程组。然后本文开发了两种使用简单$C^0$有限元的最小二乘有限元方法，一种是使用线性元的$L^2$-LSFEM，另一种是具有网格相关权重的加权 LSFEM，以确保最佳收敛。在 PDE 具有唯一解的非常温和的假设下，证明了最优先验和后验误差估计。使用比传统假设弱的算子正则性的额外假设，还讨论了加权 LSFEM 标准范数的收敛性。$L^2$-误差估计是针对两种公式得出的。我们对平滑和奇异解的平滑、非平滑甚至退化系数进行了广泛的数值实验，以测试所提出方法的准确性和效率。使用比传统假设弱的算子正则性的额外假设，还讨论了加权 LSFEM 标准范数的收敛性。$L^2$-误差估计是针对两种公式得出的。我们对平滑和奇异解的平滑、非平滑甚至退化系数进行了广泛的数值实验，以测试所提出方法的准确性和效率。使用比传统假设弱的算子正则性的额外假设，还讨论了加权 LSFEM 标准范数的收敛性。$L^2$-误差估计是针对两种公式得出的。我们对平滑和奇异解的平滑、非平滑甚至退化系数进行了广泛的数值实验，以测试所提出方法的准确性和效率。