We derive upper and lower a posteriori estimates for the maximum norm error in finite element solutions of monotone semi-linear equations. The estimates hold for Lagrange elements of any fixed order, non-smooth nonlinearities, and take numerical integration into account. The proof hinges on constructing continuous barrier functions by correcting the discrete solution appropriately, and then app...

In this paper we introduce and analyze an a posteriori error estimator for the linear finite element approximations of the Steklov eigenvalue problem. We define an error estimator of the residual type which can be computed locally from the approximate eigenpair and we prove that, up to higher order terms, the estimator is equivalent to the energy norm of the error. Finally, we prove that the vo...

Abstract. For elliptic interface problems in two and three dimensions, this paper studies a priori and residual-based a posteriori error estimations for the Crouzeix–Raviart nonconforming and the discontinuous Galerkin finite element approximations. It is shown that both the a priori and the a posteriori error estimates are robust with respect to the diffusion coefficient, i.e., constants in th...

In this paper, we derive a posteriori error estimates for space-discrete approximations of the timedependent Stokes equations. By using an appropriate Stokes reconstruction operator, we are able to write an auxiliary error equation, in pointwise form, that satisfies the exact divergence-free condition. Thus, standard energy estimates from partial differential equation theory can be applied dire...

Abstract. This two-part paper examines two approaches for mesh adaptation, using combinations of a posteriori error estimates and domain decomposition. In the first part, we consider a domain decomposition method applied to the generalized Stokes problem, with mesh adaptation in each subdomain using the a posteriori local error estimator as adaptation indicator. We apply domain decomposition wi...

Some aspects of goal-oriented a posteriori error estimation are addressed in the context of steady convection-diffusion equations. The difference between the exact and approximate values of a linear target functional is expressed in terms of integrals that depend on the solutions to the primal and dual problems. Gradient averaging techniques are employed to separate the element residual and dif...

A posteriori error estimates are derived in the context of two-dimensional structural elastic shape optimization under the compliance objective. It is known that the optimal shape features are microstructures that can be constructed using sequential lamination. The descriptive parameters explicitly depend on the stress. To derive error estimates the dual weighted residual approach for control p...

The paper is devoted to complementary approaches in a posteriori error estimation for a diffusion-reaction model problem. These approaches provide sharp and guaranteed upper bounds for the energy norm of the error and they are independent from the way how the approximate solution is obtained. In particular, the estimator naturally includes all sources of errors of any conforming approximation, ...

We give an a posteriori error estimator for nonconforming finite element approximations of diffusionreaction and Stokes problems, which relies on the solution of local problems on stars. It is proved to be equivalent to the energy error up to a data oscillation, without requiring Helmholtz decomposition of the error nor saturation assumption. Numerical experiments illustrate the good behavior a...

We present functional-type a posteriori error estimates in isogeometric analysis (IGA). These estimates, derived on functional grounds, provide guaranteed and sharp upper bounds of the true error in the energy norm. By exploiting the properties of non-uniform rational B-splines (NURBS), we present efficient computation of these error estimates. The numerical realization and the quality of the c...