Two‐grid methods for –P1 mixed finite element approximation of general elliptic optimal control problems with low regularity

Abstract

AbstractIn this paper, we present a two‐grid mixed finite element scheme for distributed optimal control governed by general elliptic equations. –P1 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 h2 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.