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1、湘潭大学 博士学位论文 几类最优控制问题的混合元方法改超收敛研究 姓名：邢小青 申请学位级别：博士 专业：计算数学 指导教师：陈艳萍 20080508 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?(? ?)? ?Raviart-Thomas? ? ? ? ? ? ? ? ?J. Douglas? ? ? ?L2? ? ? ?V. Thom ee? ? ? ? ? ? ? ?O(h 3 2) ? ? I Abstract There have existed many works about the optimal control problems governed by part
2、ial diff erential equations. Most of these researches aim at the standard fi nite element method while there doesnt seem to exist much works on theoretical analysis of mixed fi nite element methods. Mixed fi nite element methods have many advantages. For example, in computational fl uid control prob
3、lem both the scalar variable and its fl ux variable can be approximated in the same accuracy by mixed methods. It has many signifi cance to extend the standard fi nite element method to mixed fi nite element methods for optimal control problems. In this paper, we will investigate the error estimates
4、 and superconvergence properties for some optimal control problems by mixed methods. The paper consists of two parts. In the fi rst part, we study the elliptic op- timal control problem.In the analysis, we transform a minimization problem to a coupled system of state equation, co-state equation and
5、a variational in- equality. Then, we discretize the obtained optimality condition by mixed fi nite element method and approximate the states variable (scalar and vector) and con- trol variable with diff erent fi nite element spaces. First, for quadratic and general convex functional, we investigate
6、the maximum error estimates for optimal control problem with obstacle constraint set. More importantly, we introduce a special projection operator aiming at the low regularity of control variable. Then, only for the quadratic functional, we give a priori error estimate for one class of optimal contr
7、ol problem with special constraint set. At last, some numerical experiments are proposed. In the second part, we discuss the parabolic optimal control problem. There have existed some researches from V. Thom ee for parabolic equation which dont focus on the optimal control problem. First, we study a
8、 priori error estimates for this problem by mixed methods. What follows is the superconvergence analysis under rectangulation. In these works, the diffi culty is how to use the convexity and continous diff erentiability of the objective functional. Moreover, we need to con- struct some intermediate
9、variables with which we can deompose the fi nite element error into several parts. At last, we prove that the projection of state, costate vari- ables and control variable are superclose, which is O(h 3 2), to their approximations. Keywords: optimal control; mixed fi nite element; error estimates; s
10、upercon- vergence. II ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?1,3,35,46,50,54,68,94? ?34,53,64,83,93? ? 1.1? ? ?45?49.? ? ?L2 ?H1-? ?1973?Falk?45? ?1979?Geveci?49?Neuman?Arnautu ?Neittaanm aki 5?1998?2004? Meyer?R osch 76?2005?77? ?2000?Arada?Raymond?4? ?2002?Arada, Casas? Tr oltzsch?3? ?Casas 14? ? ? ? ?62,6672,103105? ?2,33, 57,60,61,7375,94,95
