Helmholtz方程的一类最小二乘混合有限元解法

A Least-Squares Mixed Finite Element Method for the HELMHOLTZ Equation

通过将Helmholtz方程变化为一阶线性系统,并考虑此线性系统余量与真解的关系,给出了对方程的一类最小二乘混合有限元方法。最小二乘混合元方法可以避免标准混合元格式中的限制条件,从而可以在更广泛的范围内选择有限元空空间。文章提出了解决问题的有限元格式,证明了离散解的存在性唯一性,并给出了误差的H(div),H1模估计。

This paper develops a mixed least-squares finite element method for the Helmholtz equation through casting the equation into a first order system, then considering the fact that the true solution minimize the remains of the system. A mixed LSM could skip the conditions which are required for a standard mixed method, so we could choose appropriate finite element spaces more freely. This paper gives the finite element formula and the proof of the existence and uniqueness of the discrete solution. Error estimates both in H（div）, H^1 norm are achieved.