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In two and three dimensional Lipschitz, but not necessarily convex, polytopal domains, we devise and analyze a reliable and efficient a posteriori error estimator for a semilinear optimal control problem; control constraints are also considered. We consider a fully discrete scheme that discretizes the state and adjoint equations with piecewise linear functions and the control variable with piecewise constant functions. The devised error estimator can be decomposed as the sum of three contributions which are associated to the discretization of the state and adjoint equations and the control variable. We extend our results to a scheme that approximates the control variable with piecewise linear functions and also to a scheme that approximates the solution to a nondifferentiable optimal control problem. We illustrate the theory with two and three-dimensional numerical examples.

This work is devoted to the finite element discretization of the incompressible Navier--Stokes equations. The starting point is a low order stabilized finite element method using piecewise linear continuous discrete velocities and piecewise constant pressures. This pair of spaces needs to be stabilized, and, as such, the continuity equation is modified by adding a stabilizing bilinear form based on the jumps of the pressure. This modified continuity equation can be rewritten in a standard way involving a modified different velocity field, which is as a consequence divergence-free. This modified velocity field is then fed back to the momentum equation making the convective term skew-symmetric. Thus, the discrete problem can be proven stable without the need to rewrite the convective field in its skew-symmetric way. Error estimates with constants independent of the viscosity are proven. Numerous numerical experiments confirm the theoretical results.

We propose and analyze a reliable and efficient a posteriori error estimator for the pointwise tracking optimal control problem of the Stokes equations. This linear-quadratic optimal control proble...

This work is devoted to the proposal, analysis, and numerical testing of a fully computable a posteriori error bound for a class of nonlinear discretizations of the convection-diffusion-reaction equation. The type of discretization we consider is nonlinear, since it has been built with the aim of preserving the discrete maximum principle. Under mild assumptions on the stabilizing term, we obtain an a posteriori error estimator that provides a certified upper bound on the norm of the error. Under the additional assumption that the stabilizing term is both Lipschitz continuous and linearity preserving, the estimator is shown to be locally efficient. We present examples of discretizations that satisfy these two requirements, and the theory is illustrated by several numerical experiments in two and three space dimensions.

The purpose of this work is the design and analysis of a posteriori error estimators for low-order stabilized finite element approximations of a generalized Boussinesq problem. We consider standard stabilization procedures over conforming finite element spaces and a nonconforming one that delivers a divergence-free discrete velocity field. The analysis, that is valid for two and three-dimensional domains, relies on a smallness assumption on the solution and is based on a technique that involves the Ritz projection of the residuals. The devised a posteriori error estimators are proven to be globally reliable and locally efficient. Three dimensional numerical experiments reveal a competitive performance of adaptive procedures driven by the designed a posteriori error estimators.

In this work we propose and analyze a new stabilized finite element method for the coupled Navier–Stokes/temperature (or Boussinesq) equations. The method is built using low order conforming elements for velocity and temperature, and piecewise constant elements for pressure. With the help of the lowest order Raviart–Thomas space, a lifting of the jumps of the discrete pressure is built in such a way that when this lifting is added to the conforming velocity field, the resulting velocity is solenoidal (at the price of being non-conforming). This field is then fed to the momentum and temperature equations, guaranteeing that the convective terms in these equations are antisymmetric, without the need of altering them, thus simplifying the analysis of the resulting method. Existence of solutions, discrete stability, and optimal convergence are proved for both the conforming velocity field, and its corresponding divergence-free non-conforming counterpart. Numerical results confirm the theoretical findings, as well as the gain provided by the solenoidal discrete velocity field over the conforming one.

We propose and analyze a reliable and efficienta posteriorierror estimator for a control-constrained linear-quadratic optimal control problem involving Dirac measures; the control variable corresponds to the amplitude of forces modeled as point sources. The proposeda posteriorierror estimator is defined as the sum of two contributions, which are associated with the state and adjoint equations. The estimator associated with the state equation is based on Muckenhoupt weighted Sobolev spaces, while the one associated with the adjoint is in the maximum norm and allows for unbounded right hand sides. The analysis is valid for two and three-dimensional domains. On the basis of the deviseda posteriorierror estimator, we design a simple adaptive strategy that yields optimal rates of convergence for the numerical examples that we perform.

We derive globally reliable a posteriori error estimators for a linear-quadratic optimal control problem involving the generalized Oseen equations as state equations; control constraints are also considered. The corresponding local error indicators are locally efficient. The assumptions under which we perform the analysis are such that they can be satisfied for a wide variety of stabilized finite element methods as well as for standard finite element methods. When stabilized methods are considered, no a priori relation between the stabilization terms for the state and adjoint equations is required. If a lower bound for the inf-sup constant is available, a posteriori error estimators that are fully computable and provide guaranteed upper bounds on the norm of the error can be obtained. We illustrate the theory with numerical examples.

We propose a posteriori error estimators for classical low-order inf–sup stable and stabilized finite element approximations of the Stokes problem with singular sources in two and three dimensional Lipschitz , but not necessarily convex, polytopal domains. The designed error estimators are proven to be reliable and locally efficient. On the basis of these estimators we design a simple adaptive strategy that yields optimal rates of convergence for the numerical examples that we perform.

We propose and analyse reliable and efficient a posteriori error estimators for an optimal control problem that involves a nondifferentiable cost functional, the Poisson problem as state equation and control constraints. To approximate the solution to the state and adjoint equations we consider a piecewise linear finite element method, whereas three different strategies are used to approximate the control variable: piecewise constant discretization, piecewise linear discretization and the so-called variational discretization approach. For the first two aforementioned solution techniques we devise an error estimator that can be decomposed as the sum of four contributions: two contributions that account for the discretization of the control variable and the associated subgradient and two contributions related to the discretization of the state and adjoint equations. The error estimator for the variational discretization approach is decomposed only in two contributions that are related to the discretization of the state and adjoint equations. On the basis of the devised a posteriori error estimators, we design simple adaptive strategies that yield optimal rates of convergence for the numerical examples that we perform.

We derive a posteriori error estimators for an optimal control problem governed by a convection–reaction–diffusion equation; control constraints are also considered. We consider a family of low-order stabilized finite element methods to approximate the solutions to the state and adjoint equations. We obtain a fully computable a posteriori error estimator for the optimal control problem. All the constants that appear in the upper bound for the error are fully specified. Therefore, the proposed estimator can be used as a stopping criterion in adaptive algorithms. We also obtain a robust a posteriori error estimator for when the error is measured in a norm that involves the dual norm of the convective derivative. Numerical examples, in two and three dimensions, are presented to illustrate the theory.

We propose computable a posteriori error estimates for a second order nonconforming finite element approximation of the Stokes problem. The estimator is completely free of unknown constants and gives a guaranteed numerical upper bound on the error, in terms of a lower bound for the inf-sup constant of the underlying continuous problem. The estimator is also shown to provide a lower bound on the error up to a constant and higher order data oscillation terms. Numerical results are presented illustrating the theory and the performance of the estimator.

The purpose of this work is the design and analysis of a reliable and efficient a posteriori error estimator for the so-called pointwise tracking optimal control problem. This linear-quadratic optimal control problem entails the minimization of a cost functional that involves point evaluations of the state, thus leading to an adjoint problem with Dirac measures on the right hand side; control constraints are also considered. The proposed error estimator relies on a posteriori error estimates in the maximum norm for the state and in Muckenhoupt weighted Sobolev spaces for the adjoint state. We present an analysis that is valid for two and three-dimensional domains. We conclude by presenting several numerical experiments which reveal the competitive performance of adaptive methods based on the devised error estimator.

SUMMARY Fully computable upper bounds are developed for the discretisation error measured in the natural (energy) norm for convection–reaction–diffusion problems in three dimensions. The upper bounds are genuine upper bounds in the sense that the numerical value of the estimated error exceeds the actual numerical value of the true error regardless of the coarseness of the mesh or the nature of the data for the problem. All constants appearing in the bounds are fully specified. Examples show the estimator to be reliable and accurate even in the case of complicated three-dimensional problems. Copyright © 2013 John Wiley & Sons, Ltd.

We study non-proper uniformly elliptic fully nonlinear equations involving extremal operators of Pucci type. We prove the existence of all radial spectrum for this type of operators and establish a multiplicity existence results through global bifurcation.