Analysis and finite element approximations of parabolic saddle point problems with applications to optimal control

We present some results concerning boundary optimal control problems and related initial-boundary value problems. We prove the existence and uniqueness of the solution of a parabolic saddle point problem, as well as the existence and uniqueness of a penalized and an iterated penalized saddle point problem. Moreover, we derive semidiscrete error estimates for the finite element approximation of the penalized saddle point problem, and semidiscrete error estimates for the penalized and unpenalized heat equation with nonhomogeneous boundary data under minimal regularity assumptions. Finally, we use the above results for the analysis and finite element analysis of boundary optimal control problems having states constrained to parabolic partial differential equations

A discontinuous Galerkin time-stepping scheme for the velocity tracking problem

The velocity tracking problem for the evolutionary Navier–Stokes equations in two dimensions is studied. The controls are of distributed type and are submitted to bound constraints. First and second order necessary and sufficient conditions are proved. A fully discrete scheme based on the discontinuous (in time) Galerkin approach, combined with conforming finite element subspaces in space, is proposed and analyzed. Provided that the time and space discretization parameters, τ and h, respectively, satisfy τ ≤ Ch2 , then L 2 error estimates of order O(h) are proved for the difference between the locally optimal controls and their discrete approximations.This author’s work was partially supported by the Spanish Ministerio de Economía y Competitividad under project MTM2011-2271

Bilinear control of semilinear elliptic PDEs: Convergence of a semismooth Newton method

In this paper, we carry out the analysis of the semismooth Newton method for bilinear control problems related to semilinear elliptic PDEs. We prove existence, uniqueness and regularity for the solution of the state equation, as well as differentiability properties of the control to state mapping. Then, first and second order optimality conditions are obtained. Finally, we prove the superlinear convergence of the semismooth Newton method to local solutions satisfying no-gap second order sufficient optimality conditions as well as a strict complementarity condition

Analysis of the velocity tracking control problem for the 3D evolutionary Navier-Stokes equations

The velocity tracking problem for the evolutionary Navier–Stokes equations in three dimensions is studied. The controls are of distributed type and are submitted to bound constraints. The classical cost functional is modified so that a full analysis of the control problem is possible. First and second order necessary and sufficient optimality conditions are proved. A fully discrete scheme based on a discontinuous (in time) Galerkin approach, combined with conforming finite element subspaces in space, is proposed and analyzed. Provided that the time and space discretization parameters, τ and h, respectively, satisfy τ ≤ Ch2, the L2(ΩT ) error estimates of order O(h) are proved for the difference between the locally optimal controls and their discrete approximations. Finally, combining these techniques and the approach of Casas, Herzog, and Wachsmuth [SIAM J. Optim., 22 (2012), pp. 795–820], we extend our results to the case of L1(ΩT ) type functionals that allow sparse controls.This author was partially supported by the Spanish Ministerio de Economía y Competitividad under projects MTM2011-22711 and MTM2014-57531-

Error estimates for the approximation of the velocity tracking problem with Bang-Bang controls

The velocity tracking problem for the evolutionary Navier–Stokes equations in 2d is studied. The controls are of distributed type but the cost functional does not involve the usual quadratic term for the control. As a consequence the resulting controls can be of bang-bang type. First and second order necessary and sufficient conditions are proved. A fully-discrete scheme based on discontinuous (in time) Galerkin approach combined with conforming finite element subspaces in space, is proposed and analyzed. Provided that the time and space discretization parameters, τ and h respectively, satisfy τ ≤ Ch2 , then L2 error estimates are proved for the difference between the states corresponding to locally optimal controls and their discrete approximations.The first author was partially supported by the Spanish Ministerio de Economía y Competitividad under projects MTM2011-22711 and MTM2014-57531-P

Error estimates for the discretization of the velocity tracking problem

In this paper we are continuing our work (Casas and Chrysafinos, SIAM J Numer Anal 50(5):2281–2306, 2012), concerning a priori error estimates for the velocity tracking of two-dimensional evolutionary Navier–Stokes flows. The controls are of distributed type, and subject to point-wise control constraints. The discretization scheme of the state and adjoint equations is based on a discontinuous time-stepping scheme (in time) combined with conforming finite elements (in space) for the velocity and pressure. Provided that the time and space discretization parameters, t and h respectively, satisfy t = Ch2, error estimates of order O(h2) and O(h 3/2 – 2/p ) with p > 3 depending on the regularity of the target and the initial velocity, are proved for the difference between the locally optimal controls and their discrete approximations, when the controls are discretized by the variational discretization approach and by using piecewise-linear functions in space respectively. Both results are based on new duality arguments for the evolutionary Navier–Stokes equations

Approximations of parabolic integro-differential equations using wavelet-Galerkin compression techniques

Abstract. Error estimates for Galerkin discretizations of parabolic integro-differential equations are presented under minimal regularity assumptions. The analysis is applicable in case that the full Galerkin matrix A associated to the integral operator is replaced by a compressed "sparse" matrixà using wavelet basis techniques. In particular, a semidiscrete (in space) scheme and a fully-discrete scheme which is discontinuous in time but conforming in space are analyzed. AMS subject classification (2000): 65R20, 65M60

Analysis and finite element approximations of parabolic saddle point problems with applications to optimal control

We present some results concerning boundary optimal control problems and related initial-boundary value problems. We prove the existence and uniqueness of the solution of a parabolic saddle point problem, as well as the existence and uniqueness of a penalized and an iterated penalized saddle point problem. Moreover, we derive semidiscrete error estimates for the finite element approximation of the penalized saddle point problem, and semidiscrete error estimates for the penalized and unpenalized heat equation with nonhomogeneous boundary data under minimal regularity assumptions. Finally, we use the above results for the analysis and finite element analysis of boundary optimal control problems having states constrained to parabolic partial differential equations.</p

Stability analysis and best approximation error estimates of discontinuous time-stepping schemes for the Allen–Cahn equation

Fully-discrete approximations of the Allen–Cahn equation are considered. In particular, we consider schemes of arbitrary order based on a discontinuous Galerkin (in time) approach combined with standard conforming finite elements (in space). We prove that these schemes are unconditionally stable under minimal regularity assumptions on the given data. We also prove best approximation a-priori error estimates, with constants depending polynomially upon (1/ε) by circumventing Gronwall Lemma arguments. The key feature of our approach is a carefully constructed duality argument, combined with a boot-strap technique