Least-squares finite element for second order boundary value problem with optimal rates of convergence

摘要

During the past twenty years, engineers and mathematicians have successfully developed the mixed methods and the hybrid methods to solve problems (1.1), (1.2) and (1.3) numerically, such as [1–6]. The discrete inf-sup or its equivalent condition for the subspaces of the approximate functions of u and p has been applied for. It ensures the numerical results converge to the real world solution steadily. In this paper, we try to put an extra equation which is the compatibility condition of this system. We then apply the least-squares method to the so-called enriched system. Analysis shows that this method achieves the optimal rate of convergence in the H1and L2 norms without using the above condition. Numerical examples are still available.