Numerical approximation of control problems of non-monotone and non-coercive semilinear elliptic equations

Abstract

We analyze the numerical approximation of a control problem governed by a non-monotone and non-coercive semilinear elliptic equation. The lack of monotonicity and coercivity is due to the presence of a convection term. First, we study the finite element approximation of the partial differential equation. While we can prove existence of a solution for the discrete equation when the discretization parameter is small enough, the uniqueness is an open problem for us if the nonlinearity is not globally Lipschitz. Nevertheless, we prove the existence and uniqueness of a sequence of solutions bounded in $$L^\infty (\varOmega )$$ L ∞ ( Ω ) and converging to the solution of the continuous problem. Error estimates for these solutions are obtained. Next, we discretize the control problem. Existence of discrete optimal controls is proved, as well as their convergence to solutions of the continuous problem. The analysis of error estimates is quite involved due to the possible non-uniqueness of the discrete state for a given control. To overcome this difficulty we define an appropriate discrete control-to-state mapping in a neighbourhood of a strict solution of the continuous control problem. This allows us to introduce a reduced functional and obtain first order optimality conditions as well as error estimates. Some numerical experiments are included to illustrate the theoretical results.

A Priori and a Posteriori Error Analysis for Generic Linear Elliptic Problems

“…In this section, we consider deriving a posteriori error estimate for the general linear system of elliptic equations in (30) ).…”

“…Ye and Zhang in [29] analysed and studied the error estimates for continuous and discontinuous weak Galerkin (WG) FEMs for elliptic problems with low regularity solutions in energy and 𝐿 2 norms. Casas and coworkers [30] examined the numerical solution of semilinear elliptic equations. They proved the existence and uniqueness of a sequence of bounded solutions in 𝐿 ∞ (Ω).…”

“…which is a vector of finite element approximations of the functions u and v. Hence the problem becomes: find 𝑤 ℎ ∈ 𝑉 ℎ such that 𝑎(𝑤 ℎ , 𝜑) = (𝜑), ∀𝜑 ∈ 𝑉 ℎ . (35) Theorem 3.2 (𝑯 𝟎 𝟏 A Priori Error Bound for a Generic Linear Elliptic System) The finite element approximate solution w h of the problem(30), satisfies the following a priori energy (H 0 1 ) error estimate||e|| 0 = ||w − w h || 0 ≤ 𝐶 ̃ℎ ̃(||u|| L2(Ω) + ||v|| L2(Ω) ) = 𝐶 ̃ℎ ̃||Dw|| L2(Ω) . (36) Proof.…”