Full-discrete adaptive FEM for quasi-parabolic integro-differential PDE-constrained optimal control problem

In many applications such as mechanics, heat conduction in materials with memory, and so on, models are often described by quasi-parabolic integro-differential PDEs (IDPDEs). In any subdomain, these models maintain conservation of energy at any moment. In [1], the authors presented the existence and uniqueness of the solution to a quasi-parabolic IDPDE. In [2], the authors studied the finite element methods (FEMs) for quasi-parabolic IDPDE problems.

For classic linear PDE-constrained optimal control problems (OCPs), such as elliptic, parabolic, and hyperbolic equation constrained OCPs. there has been much work on the existence and optimality conditions (see [3]). In [4–11], the authors have already carefully studied the FEM approximation and obtained a priori error estimates. Recently, in [12, 13], the authors concentrated themselves on OCPs governed by IDPEDs such as elliptic, parabolic, and quasi-parabolic IDPDEs. However, although we often encounter those control problems in many practical applications, as far as we know, there has been little study on full-discrete adaptive finite element methods (AFEMs) for quasi-parabolic IDPDE constrained OCP.

As is well known, AFEM can efficiently decrease the computational work among different FEMs in the literature, and it has become one of the main methods in developing FEMs in real scientific and engineering applications. The principle of this method is to use the a posteriori error estimate indicator to guide local mesh refinement in computation; in such a way denser meshes are only used where the value of the indicator is larger and the solution is more singular, and consequently the degree of freedoms may be greatly reduced on those only locally refined meshes. One of the keys to make AFEM work is that the value of the error estimate indicator has to be highly positively correlated to the singularity of the solutions to be approximated. So the quality of the a posteriori error indicators is vital for effectiveness of AFEM [14]. In [15–18], the authors present the effectiveness of AFEM and carry out tests to illustrate that AFEM can indeed reduce computational work for PDE-constrained OCP.

In this paper, the a posteriori error indicators are presented for AFEM approximation of (1.1)-(1.2). For this control problem, by carrying out a very detailed analysis, we obtain the a posteriori error estimates for the full-discrete AFEM on multimeshes. Here we also employ and extend the existing techniques in [12, 13, 19–21] to (1.1)-(1.2).

The outline of the paper is as follows. In Section 2, we present the weak formulation and optimal conditions for (1.1)-(1.2). In Section 3, the semidiscrete and full-discrete FEM approximation forms for (1.1)-(1.2) are established. In Section 4, we derive upper and lower a posteriori error bounds for (1.1)-(1.2). Finally, we carry out some numerical experiments to show the effectiveness of the indicators derived in Section 4.

In this paper, Ω is a bounded open set with Lipschitz boundary ∂Ω, and \(\Omega_{U}\) is another bounded open set in \(R^{d}\), where \(1 \leq d \leq3\).

In the following, we use the standard notation \(W^{m,q}(\Omega)\) for Sobolev spaces on Ω equipped with norm \(\|\cdot\|_{m,q,\Omega}\), and seminorm \(|\cdot|_{m,q,\Omega}\). We denote \(W_{0}^{m,q}(\Omega)= \{w \in W^{m,q}(\Omega) : w|_{\partial\Omega} = 0 \}\). For convenience, we use \(H^{m}(\Omega)\) \((H_{0}^{m}(\Omega))\) to denote \(W^{m,2}(\Omega)\) \((W_{0}^{m,2}(\Omega)) \) equipped with norm\(\|\cdot\|_{m,\Omega}\) and seminorm \(|\cdot |_{m,\Omega}\). The Banach space of all \(L^{s}\) integrable functions (\((0,S)\rightarrow W^{m,q}(\Omega)\)) is denoted by \(L^{s} (0,S;W^{m,q}(\Omega)) \), which is equipped with norm \(\| v\|_{L^{s}(0,S;W^{m,q}(\Omega))} = (\int_{0}^{S}{\| v \|^{s}_{W^{m,q}(\Omega)}\,dt})^{\frac{1}{s}} \) for \(s \in[1,\infty) \) and with the standard modification for \(s=\infty\). In the same way, we also define the spaces \(H^{1}(0,S;W^{m,q}(\Omega))\) and \(C^{k}(0,S;W^{m,q}(\Omega))\). In [22], we can find the details.

Throughout the paper, we denote by c or C general positive constants independent of any unknowns and mesh parameters.

Let \(V=H^{1}_{0}(\Omega)\) and \(U=L^{2}(\Omega)\); as the state and control spaces, we take \(W =L^{2}(0,T; V)\) and \(X= L^{2}(0,T;U)\).

Also, we let \(K= \{u(t) \in L^{2} (\Omega): \int_{\Omega}u(t) \geq0, \forall t\in[0,T]\}\) be a closed convex subset of U, and as the control set, we take \(U_{ad}=\{u \in X, u(t)\in K\}\).

We now consider the full-discrete FEM approximation to (1.1)-(1.2). For simplicity, here we only consider triangular conforming elements and suppose that \(\Omega^{h}=\Omega\).

Let \(T^{h}\) and \(T_{U}^{h}\) be two different polygonal approximations to \(\Omega^{h}\). On \(T^{h}\), we set the finite element subset \(S^{h}\) of \(C(\bar{\Omega}^{h})\) that satisfies: \(\forall \chi\in S^{h}\), \(\forall \tau\in T^{h} \), \(\chi|_{\tau}\) is an m-polynomial, where \(m\ge1\) is the degree of the polynomials. Let \(V^{h}=\{v_{h}\in S^{h}: v_{h}(P_{i})=0, i=1,\ldots, J\}\) and \(W^{h}=L^{2}(0,T;V^{h})\), where \(P_{i}\) is the boundary vertex of the domain \(T^{h}\) (\(i=1,\ldots, J\)). Obviously, we have \(V^{h}\subset V\), \(W^{h}\subset W\). On \(T_{U}^{h}\), we set the finite element subset \(U^{h}\) in \(L^{2}({\Omega}^{h})\) that satisfies: \(\forall \chi\in U^{h}\), \(\forall \tau_{U}\in T_{U}^{h} \), \(\chi|_{\tau_{U}}\) is an m-polynomial (\(m \geq1\)). Let \(X^{h}=L^{2}(0,T;U^{h})\). Note that we do not require the continuity for \(U^{h}\). Then we obviously have \(X^{h}\subset X\).

Let the diameter of the maximal circumcircle of an element τ (\(\tau _{U}\)) in \(T^{h} \) (\(T^{h}_{U}\)) be denoted by \(h_{\tau}\) (\(h_{\tau_{U}}\)), and the diameter of the maximal inscribed circle of element τ (\(\tau _{U}\)) in \(T^{h} \) (\(T^{h}_{U}\)) be denoted by \(\rho_{\tau}(\rho_{\tau_{U}} )\). Suppose that there exists a positive constant R such that \(1\leq\max_{\tau\in T^{h}}(\frac{h_{\tau}}{\rho_{\tau}})\leq R\) and \(1\leq\max_{\tau_{U}\in T_{U}^{h}}(\frac{h_{\tau_{U}}}{\rho_{\tau_{U}}})\leq R \), and let \(h=\max_{\tau\in T^{h}}h_{\tau}\) and \(h_{U}=\max_{\tau_{U}\in T_{U}^{h}}h_{\tau_{U}}\).

Here piecewise constant FE space is used for the approximation of the control due to the limited regularity of the optimal control (normally, \(H^{1}\)). We will use higher-order FE spaces when approximating the state and costate. We denote all 0-order polynomials on \({\tau}_{U}\) by \(P_{0}({\tau }_{U})\) and always set \(X^{h} = \{u \in X: u(x, t)| _{x\in\tau_{U}} \in P_{0}(\tau_{U}), \forall t \in [0,T]\}\) and \(K^{h}=\{u_{h}(t)\in U^{h}: \int_{{\Omega}} u_{h}(t) \geq 0\}\).

Now we use the backward Euler scheme in time to construct the full-discrete FEM approximation of (3.3) and obtain its full-discrete form. For convenience, we suppose that Ω is a polygonal area with boundary ∂Ω.

Let \(0 = t_{0} < t_{1} <\cdots< t_{N} = T\), \(k_{i} = t_{i}-t_{i-1}\), \(i = 1, 2,\ldots, N\), \(k = \max_{1\leq i\leq N}\{ k_{i}\}\). For \(i = 1, 2,\ldots, N\), we similarly to \(V^{h}\) construct the FE space \(V^{h}_{i}\subset H_{0}^{1}(\Omega)\) on mesh \(T^{h}_{i}\). We also construct the FE space \(U^{h}_{i}\subset L^{2}(\Omega)\) on mesh \((T^{h}_{U})_{i}\). Here we denote the maximal diameter of the element \({\tau^{i}}\) (\({\tau_{U}^{i}}\)) in \(T_{i}^{h}\) (\((T_{U}^{h})_{i}\)) by \(h_{\tau^{i}}\) (\(h_{\tau_{U}^{i}}\)). Then we define the mesh functions \(\tau(\cdot)\) and \(\tau_{U}(\cdot)\) by \(\tau(t)|_{t\in (t_{i-1}, t_{i}]}=\tau^{i}\) and \(\tau_{U}(t)|_{t\in(t_{i-1}, t_{i}]}=\tau_{U}^{i}\). We also define the mesh size functions \(h_{\tau}(\cdot)\) and \(h_{\tau_{U}}(\cdot)\) by \(h_{\tau}(t)|_{t\in (t_{i-1}, t_{i}]}=h_{\tau^{i}}\) and \(h_{\tau_{U}}(t)|_{t\in(t_{i-1}, t_{i}]}=h_{\tau_{U}^{i}}\). In the following, for convenience, we use τ, \(\tau_{U}\), \(h_{\tau}\), and \(h_{\tau_{U}}\) to denote \(\tau(t)\), \(\tau_{U}(t)\), \(h_{\tau}(t)\), and \(h_{\tau_{U}}(t)\), respectively. Let \(K_{i}^{h} \subseteq U^{h}_{i}\cap K \).

In order to present the full-discrete FEM approximation of \((OCP)^{h}\), we set \(t=t_{i}\) in the first formula of (3.2), and in order to approximate derivative in time, we adopt the 1-order backward difference quotient. We also use the left rectangular quadrature formula to numerically approximate the integration.

Since there is huge computational work in computing \((OCP\mbox{-}OPT)^{hk}\), we need to develop a highly efficient algorithm to compute \((OCP\mbox{-}OPT)^{hk}\). Here different partitions of meshes are used for the state and control.

First, we present the following two lemmas. Lemma 4.1 can be found in [24], and Lemma 4.2 can be found, for example, in Chapter 3 of [25].

In this section, we carry out a numerical experiment to verify the a posteriori error estimates derived in Section 4. Let \(\Omega=\Omega_{U} = (0,1)^{2}\). In order to approximate the state and costate, we use linear FE spaces, and similarly to approximate the control, we use the piecewise constant finite element spaces. We also choose the Euler backward-difference procedure to compute the full-discrete system for time variable.

Here we use the AFEpack software package introduced in [27] to conduct our numerical experiment. The experiment is conducted on a uniform mesh and an adaptive mesh. We use the same time step \(dt =0.05\) for u, p, y. The initial mesh for adaptive mesh is shown in Figure 1, and the tolerance for the adaptive mesh is 10⁻⁶. We have used the estimators \(\eta_{1}\), \(\eta_{2}\), \(\eta_{3}\), \(\eta_{5}\), \(\eta_{7}\), \(\eta_{9}\), \(\eta_{11}\), whereas the others are higher-order terms, so omitted traditionally. We use \(\eta_{1}\) for adaptive control and use \(\eta_{2}\), \(\eta_{3}\), \(\eta_{5}\), \(\eta_{7}\), \(\eta_{9}\), \(\eta_{11}\) for state and costate adaptive, respectively.

All data of the error estimates we obtain in the following tables are in norms of \(L^{2}(0,T;L^{2}(\Omega))\), \(L^{2}(0,T;H^{1}(\Omega))\), and \(L^{\infty}(0,T;L^{2}(\Omega))\). We note that, on the adaptive mesh, fewer nodes, sides, elements, and DOFs in the state variables are used. Since repeatedly solving the state and the costate equations is the main computational work in computing the control problem, we can save much computation work by using the adaptive mesh.

In Figures 2-4, we present the adaptive mesh for the control, state, and costate at different times.