Abstract
In this paper, a semidiscrete finite element method for solving bilinear parabolic optimal control problems is considered. Firstly, we present a finite element approximation of the model problem. Secondly, we bring in some important intermediate variables and their error estimates. Thirdly, we derive a priori error estimates of the approximation scheme. Finally, we obtain the superconvergence between the semidiscrete finite element solutions and projections of the exact solutions. A numerical example is presented to verify our theoretical results.
Highlights
We consider the following bilinear parabolic optimal control problem:T min u∈K y(t, x) – yd(t, x) L ( )+ u(t, x) ) dt, ( . )∂ty(t, x) – div A(x)∇y(t, x) + u(t, x)y(t, x) = f (t, x), t ∈ J, x ∈, y(t, x) =, t ∈ J, x ∈ ∂, y(, x) = y (x), x ∈, where ∈ R is a convex polygon with the boundary ∂, and J = [, T] ( < T < +∞)
There has been a wide range of research on finite element approximation of elliptic optimal control problems
6 Numerical experiment we present a numerical example to validate our superconvergence results
Summary
We consider the following bilinear parabolic optimal control problem:. ∂ty(t, x) – div A(x)∇y(t, x) + u(t, x)y(t, x) = f (t, x), t ∈ J, x ∈ , y(t, x) = , t ∈ J, x ∈ ∂ , y( , x) = y (x), x ∈ , where ∈ R is a convex polygon with the boundary ∂ , and J = [ , T] ( < T < +∞). For finite element solving linear and semilinear elliptic control problems, a priori error estimates were investigated in [ ] and [ ], and superconvergence. Yang et al [ ] obtained the superconvergence of finite element approximation of bilinear elliptic control problems. Some similar results of mixed finite element approximation for linear elliptic control problems can be found in [ , ]. A priori error estimates of space-time finite element and standard finite element approximation for linear parabolic control problem were derived in [ ] and [ ]. The superconvergence of variational discretization and standard finite element approximation for semilinear parabolic control problem can be found in [ ] and [ ], respectively. We purpose to obtain the superconvergence properties of semidiscrete finite element method for bilinear parabolic optimal control problems. A numerical example is presented to illustrate our theoretical results in the last section
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