This paper studies adaptive least-squares finite element methods for convection-dominated diffusion-reaction problems. The least-squares methods are based on the first-order system of the primal and dual variables with various ways of imposing outflow boundary conditions. The coercivity of the homogeneous least-squares functionals are established, and the a priori error estimates of the least-squares methods are obtained in a norm that incorporates the streamline derivative. All methods have the same convergence rate provided that meshes in the layer regions are fine enough. To increase computational accuracy and reduce computational cost, adaptive least-squares methods are implemented and numerical results are presented for some test problems.

翻译：本文研究求解对流主导扩散反应问题的自适应最小二乘有限元方法。最小二乘法基于原始变量与对偶变量的一阶系统，并采用多种方式施加出流边界条件。建立了齐次最小二乘泛函的强制性质，并在包含流线导数的范数下获得了最小二乘方法的先验误差估计。只要层区域中的网格足够精细，所有方法均具有相同的收敛速度。为提高计算精度并降低计算成本，实施了自适应最小二乘方法，并给出了若干测试问题的数值结果。