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Two‐grid methods for P 0 2 – P 1 mixed finite element approximation of general elliptic optimal control problems with low regularity
Author(s) -
Hou Tianliang,
Leng Haitao,
Luan Tian
Publication year - 2020
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22471
Subject(s) - mathematics , superconvergence , discretization , finite element method , piecewise , grid , optimal control , projection (relational algebra) , mixed finite element method , elliptic operator , elliptic curve , mathematical optimization , mathematical analysis , geometry , algorithm , physics , thermodynamics
In this paper, we present a two‐grid mixed finite element scheme for distributed optimal control governed by general elliptic equations.P 0 2 – P 1 mixed finite elements are used for the discretization of the state and co‐state variables, whereas piecewise constant function is used to approximate the control variable. We first use a new approach to obtain the superclose property between the centroid interpolation and the numerical solution of the optimal control u with order h 2 under the low regularity. Based on the superclose property, we derive the optimal a priori error estimates. Then, using a postprocessing projection operator, we get a second‐order superconvergent result for the control u . Next, we construct a two‐grid mixed finite element scheme and analyze a priori error estimates. In the two‐grid scheme, the solution of the elliptic optimal control problem on a fine grid is reduced to the solution of the elliptic optimal control problem on a much coarser grid and the solution of a linear algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy. Finally, a numerical example is presented to verify the theoretical results.