半线性椭圆最优控制问题插值系数混合有限元解的先验误差估计

Interpolation Coefficients Mixed Finite Element Solutions For Semilinear Elliptic Optimal Control Problems Priori Error Estimates

利用插值系数混合有限元方法求解半线性最优控制问题,采用插值系数的思想去处理方程中的非线性项,建立了半线性椭圆最优控制问题插值系数混合有限元的离散格式,将状态方程和对偶状态方程利用低阶的Raviart-Thomas混合有限元空间离散,控制变量利用分片常函数逼近,最后获得状态变量和控制变量的L2范数和H(div)范数的最优阶先验误差估计.

In this paper,the authors extend the excellent idea of interpolation coefficients for semilinear optimal control problems to the mixed finite element methods. By using the interpolation coefficients thought to process the nonlinear term of equations ,we present the mixed finite element approximation with interpolation coefficients for general optimal control problems governed by semilinear elliptic equations. The state and the co-state are discretized by the lowest order Raviart-Thomas mixed finite element space and the control is discretized by piecewise constant elements. We derive a priori error estimates in L2 norm and H （div） norm for the coupled state and control variables with optimal convergence order h2.