The Gradient Discretization Method for Optimal Control Problems, with Superconvergence for Nonconforming Finite Elements and Mixed-Hybrid Mimetic Finite Differences

Abstract

In this paper, optimal control problems governed by diffusion equations with Dirichlet and Neumann boundary conditions are investigated in the framework of the gradient discretisation method. Gradient schemes are defined for the optimality system of the control problem. Error estimates for state, adjoint and control variables are derived. Superconvergence results for gradient schemes under realistic regularity assumptions on the exact solution is discussed. These super-convergence results are shown to apply to non-conforming P 1 finite elements, and to the mixed/hybrid mimetic finite differences. Results of numerical experiments are demonstrated for the conforming, non-conforming and mixed-hybrid mimetic finite difference schemes.

Numerical Analysis for the Pure Neumann Control Problem Using the Gradient Discretisation Method

“…We use three specific schemes for the state and adjoint variables: conforming finite element (FE) method, non-conforming finite element (ncP 1 FE) method and hybrid mimetic mixed (HMM) method. All three are GDMs with gradient discretisations with bounds on C D , order h estimate on WS D , and satisfying assumptions (A1)-(A4), and (3.14) on quasi-uniform meshes; see [15,20] and Remarks 2.4 and 3.4. The control variable is discretised using piecewise constant functions on the corresponding meshes.…”

“…Recently, this result was extended to the GDM in [20]. Carrying out this analysis in the context of the GDM means that it readily applies to various schemes, including nonconforming P 1 finite elements and hybrid mimetic mixed schemes (HMM), which contains mixed-hybrid mimetic finite differences; and for these schemes, the analysis of [20] provides novel estimates. We refer to [8,9,10,11,12,13] for an analysis of nonlinear elliptic optimal control problems.…”

“…This relation, which is non-standard since it has to account for the zero average constraints, is the key to prove the super-convergence result for all three variables; • design a modified active set strategy algorithm for GDM that is adapted to this non-standard projection relation; • discuss some numerical results that confirm the theoretical rates of convergence for conforming, nonconforming and mimetic finite difference methods. The literature contains several contributions to numerical analysis for second order distributed optimal control problems governed by diffusion equation with Dirichlet and Neumann boundary conditions (BC) (we refer to [20,30] and the references therein). For Dirichlet BC, the super-convergence of post-processed controls for conforming finite element (FE) methods has been investigated in [30].…”

“…For Dirichlet BC, the super-convergence of post-processed controls for conforming finite element (FE) methods has been investigated in [30]. Recently, this result was extended to the GDM in [20]. Carrying out this analysis in the context of the GDM means that it readily applies to various schemes, including nonconforming P 1 finite elements and hybrid mimetic mixed schemes (HMM), which contains mixed-hybrid mimetic finite differences; and for these schemes, the analysis of [20] provides novel estimates.…”

Convergence Analysis of the mimetic Finite Difference Method for Elliptic Problems with Staggered Discretizations of Diffusion Coefficients

“…This issue affects also the discretization of linear diffusion problems where k is only a function of position, but may be discontinuous or close to zero in some parts of the domain. In the finite element (FE) and finite volume (FV) frameworks we mention the mixed finite element method [15], the 'standard' mimetic finite difference (MFD) method [13,39], the gradient scheme [29,30], the hybrid and mixed finite volumes method [28,[31][32][33], the hybrid high-order method [26,27], the mixed weak Galerkin method [45] and the mixed virtual element method [11,18]. On the other hand, finite difference methods and finite volume methods that approximate directly k∇p in the mass conservation equation do not invert the diffusion coefficient and do not suffer of this problem.…”