Adaptive fully-discrete finite element methods for nonlinear quadratic parabolic boundary optimal control
© Lu; licensee Springer. 2013
Received: 18 January 2013
Accepted: 14 March 2013
Published: 4 April 2013
The aim of this work is to study adaptive fully-discrete finite element methods for quadratic boundary optimal control problems governed by nonlinear parabolic equations. We derive a posteriori error estimates for the state and control approximation. Such estimates can be used to construct reliable adaptive finite element approximation for nonlinear quadratic parabolic boundary optimal control problems. Finally, we present a numerical example to show the theoretical results.
In this paper, we study the fully-discrete finite element approximation for quadratic boundary optimal control problems governed by nonlinear parabolic equations. Optimal control problems are very important models in engineering numerical simulation. They have various physical backgrounds in many practical applications. Finite element approximation of optimal control problems plays a very important role in the numerical methods for these problems. The finite element approximation of a linear elliptic optimal control problem is well investigated by Falk  and Geveci . The discretization for semilinear elliptic optimal control problems is discussed by Arada, Casas, and Tröltzsch in . Systematic introductions of the finite element method for optimal control problems can be found in [4–6].
As one of important kinds of optimal control problems, the boundary optimal control is widely used in scientific and engineering computing. The literature in this aspect is huge; see, e.g., [7–10]. For some quadratic boundary optimal control problems, Liu and Yan [11, 12] investigated a posteriori error estimates and adaptive finite element methods. Alt and Mackenroth  were concerned with error estimates of finite element approximations to state constrained convex parabolic boundary optimal control problems. Arada et al. discussed the numerical approximation of boundary optimal control problems governed by semilinear elliptic equations with pointwise constraints on the control in . Although a priori error estimates and a posteriori error estimates of finite element approximation are widely used in numerical simulations, they have not yet been utilized in nonlinear parabolic boundary optimal control problems.
Adaptive finite element approximation is the most important method to boost accuracy of the finite element discretization. It ensures a higher density of nodes in a certain area of the given domain, where the solution is discontinuous or more difficult to approximate, using a posteriori error indicator. A posteriori error estimates are computable quantities in terms of the discrete solution that measure the actual discrete errors without the knowledge of exact solutions. They are essential in designing algorithms for mesh which equidistribute the computational effort and optimize the computation. Recently, in [15–18], we derived a priori error estimates, a posteriori error estimates and superconvergence for optimal control problems using mixed finite element methods.
In this paper, we adopt the standard notation for Sobolev spaces on Ω with a norm given by and a semi-norm given by . We set . For , we denote , , and , . We denote by the Banach space of all integrable functions from J into with the norm for , and the standard modification for . The details can be found in .
The plan of this paper is as follows. In the next section, we present a finite element discretization for nonlinear quadratic parabolic boundary optimal control problems. A posteriori error estimates are established for the finite element approximation solutions in Section 3. In Section 4, we give a numerical example to prove the theoretical results.
2 Finite element methods for parabolic boundary optimal control
where the inner product in or is indicated by , and B is a continuous linear operator from U to .
where is the adjoint operator of B. In the rest of the paper, we shall simply write the product as whenever no confusion should be caused.
Let us consider the finite element approximation of control problem (9)-(11). Again, here we consider only n-simplex elements and conforming finite elements.
Let be a regular partition of Ω. Associated with is a finite dimensional subspace of such that are polynomials of m-order () and . It is easy to see that . Let be a partition of ∂ Ω into disjoint regular -simplices s, so that . Associated with is another finite dimensional subspace of such that are polynomials of m-order () and . Let denote the maximum diameter of the element in , , and . In addition C or c denotes a general positive constant independent of h.
where , is an approximation of .
For , we construct the finite element spaces with the mesh (similar to ). Similarly, we construct the finite element spaces with the mesh (similar to ). Let denote the maximum diameter of the element in . Define mesh functions , and mesh size functions , such that , , , . For ease of exposition, we denote , , , and by τ, s, , and , respectively.
where , is an approximation of .
Now we restate the following well-known estimates in .
where l is the edge of the element.
are bounded functions in .
3 A posteriori error estimates
In this section we obtain a posteriori error estimates for nonlinear quadratic parabolic boundary optimal control problems. Firstly, we estimate the error .
where n is the unit normal vector on outwards .
This completes the proof. □
Analogously to Theorem 3.1, we show the following estimates.
where n is the unit normal vector on outwards .
This completes the proof. □
where u and are the solutions of (66) and (67), respectively. We assume the above inequality throughout this paper.
Then it is clear that three subsets do not intersect each other, and , .
Let be the solution of (41)-(44). We establish the following error estimate, which can be proved similarly to the proofs given in .
Therefore, (69) follows from (70)-(72) and (76)-(78). □
Hence, we combine Theorems 3.1-3.3 to conclude the following.
where , and are defined in Theorems 3.1-3.3, respectively.
Finally, combining Theorems 3.1-3.3 and (84)-(85) leads to (79). □
4 Numerical example
In the example, we choose the domain and . Let Ω be partitioned into as described in Section 2. We use as the control mesh refinement indicator and - as the states and co-states.
where is the average of over T. In solving our discretized optimal control problem, we use the preconditioned projection gradient method (89)-(92) with and a fixed step size . In the numerical simulation, we use a piecewise linear finite element space for the approximation of y and p, and a piecewise constant for u.
Comparison of uniform mesh and adaptive mesh
ZL participated in the design of all the study and drafted the manuscript.
This work is supported by National Science Foundation of China (11201510), Mathematics TianYuan Special Funds of the National Natural Science Foundation of China (11126329), China Postdoctoral Science Foundation funded project (2011M500968), Natural Science Foundation Project of CQ CSTC (cstc2012jjA00003), and Natural Science Foundation of Chongqing Municipal Education Commission (KJ121113). The author expresses his thanks to the referees for their helpful suggestions, which led to improvements of the presentation.
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