Abstract
In this paper, we consider the a posteriori error estimates of the mixed finite element method for the optimal control problems governed by fourth order hyperbolic equations. The state is discretized by the order k Raviart–Thomas mixed elements and control is discretized by piecewise polynomials of degree k. We adopt the mixed elliptic reconstruction to derive the a posteriori error estimates for both the state and the control approximations.
Highlights
The finite element approximation for optimal control problems has an enormously important function in the numerical approach for these problems
Scientists have studied extensively this area; see, for example, [4, 12, 13, 21, 25]. They discussed the a priori error estimates using finite element approximations, such as [1, 16, 23], in which elliptic or parabolic problems are considered by optimal control theory
In some optimal control problems, for the objective function containing a gradient of the state variable, we use mixed finite element methods to discretize the state equation, so that the scalar variable and its flux variable can be approximated in the same accuracy; for example, see [3]
Summary
The finite element approximation for optimal control problems has an enormously important function in the numerical approach for these problems. Scientists have studied extensively this area; see, for example, [4, 12, 13, 21, 25] They discussed the a priori error estimates using finite element approximations, such as [1, 16, 23], in which elliptic or parabolic problems are considered by optimal control theory. By (2.59)–(2.62) and (2.69)–(2.72), we obtain the error equations (e5, v) – (e6, div v) = 0, ∀v ∈ V, (div e5, w) = (e4 + e8 + η4 + η8, w), ∀w ∈ W , (e7, v) – (e8, div v) = 0, ∀v ∈ V, (e6tt, w) + (div e7, w) = (η6tt + η2 + e2, w), ∀w ∈ W. For sufficiently small , using Lemmas 3.1–3.3 and (3.35), (3.53)–(3.54), we complete the proof
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