杂交混合有限元方法求解二维椭圆界面问题

The Hybrid Mixed Finite Element Method for Elliptic Interface Problem

采用杂交混合有限元(Hybrid mixed finite element)方法求解椭圆界面问题。由于扩散系数在界面上间断,椭圆界面问题的解和梯度在界面上会出现跳跃现象,针对二维椭圆界面问题,在三角网格上采取杂交混合有限元方法,该方法的特点表现在对于复杂的界面,三角网格剖分可以很好地拟合界面。当有限元近似空间取k阶多项式时,方程的解和梯度在L2范数下均会达到最优阶收敛,即k+1阶。由于线性方程组的矩阵是对称正定的,因此可以应用共轭梯度方法求解方程组。数值算例结果表明,杂交混合有限元方法对于求解强界面问题十分有效。

This paper discusses the hybrid mixed finite element method for elliptic interface problem in which the solution and gradient are discontinuous because of jump conditions across the interface.For the two-dimensional elliptic interface problem,we propose the hybrid mixed finite element method on triangulation.The properties of this method mainly reflect in the following：the triangular mesh can nicely approximate the complex interface;they provide the approximations of all the variables,including the solution and gradient,with the optimal order;the matrix of final linear algebra systems is symmetric and positive definite,so it can be numerically solved by using for example conjugate gradient method.Numerical examples demonstrate the validity of hybrid mixed finite element method for elliptic interface problems.