A two-grid $ P_0^2 $-$ P_1 $ mixed finite element scheme for semilinear elliptic optimal control problems

Abstract

<abstract><p>This paper aims to construct a two-grid mixed finite element scheme for distributed optimal control governed by semilinear elliptic equations. The state and co-state are approximated by the $ P_0^2 $-$ P_1 $ pair and the control variable is approximated by the piecewise constant functions. First, a superclose result for the control variable and a priori error estimates for all variables are obtained. Second, a two-grid $ P_0^2 $-$ P_1 $ mixed finite element algorithm is presented and the corresponding error is analyzed. In the two-grid scheme, the solution of the semilinear elliptic optimal control problem on a fine grid is reduced to the solution of the semilinear elliptic optimal control problem on a much coarser grid and the solution of a linear decoupled algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy. We find that the two-grid method achieves the same convergence property as the $ P_0^2 $-$ P_1 $ mixed finite element method if the two mesh sizes satisfy $ h = H^2 $. Finally, a numerical example demonstrating our theoretical results is presented.</p></abstract>

Numerical analysis of two-dimensional unsaturated soil water flow problems with two-grid finite element methods

“…Its main idea is to select a coarse-grid space to produce a rough approximation for the solution of nonlinear problems, and then use it as the initial guess on the fine-grid space. Later, two-grid methods combined with all kinds of spatial discretization methods are successfully extended to semilinear elliptic interface problems [25,7], semilinear elliptic optimal control problems [27,15], eigenvalue problems [31,32,29], and further developed for other applications, such as nonlinear Sobolev equations [30], Cahn-Hilliard equations [16], Stokes-Darcy problems [14,20], Darcy-Brinkman fracture models [6], nonlinear time-fractional parabolic equations [33], semilinear parabolic integrodifferential equations [17,4,26], nonlinear hyperbolic integro-differential equation [21], time-dependent Schrödinger equation [3,24,23], etc.…”