A posteriori error estimates for nonlinear problems. -estimates for finite element discretizations of elliptic equations
A posteriori error estimates for parabolic differential systems solved by the finite element method of lines
Systems of parabolic differential equations are studied in the paper. Two a posteriori error estimates for the approximate solution obtained by the finite element method of lines are presented. A statement on the rate of convergence of the approximation of error by estimator to the error is proved.
A posteriori error estimates for the D stabilized Mortar finite element method applied to the Laplace equation
We consider a non-conforming stabilized domain decomposition technique for the discretization of the three-dimensional Laplace equation. The aim is to extend the numerical analysis of residual error indicators to this model problem. Two formulations of the problem are considered and the error estimators are studied for both. In the first one, the error estimator provides upper and lower bounds for the energy norm of the mortar finite element solution whereas in the second case, it also estimates...
A posteriori Error Estimates For the 3D Stabilized Mortar Finite Element Method applied to the Laplace Equation
We consider a non-conforming stabilized domain decomposition technique for the discretization of the three-dimensional Laplace equation. The aim is to extend the numerical analysis of residual error indicators to this model problem. Two formulations of the problem are considered and the error estimators are studied for both. In the first one, the error estimator provides upper and lower bounds for the energy norm of the mortar finite element solution whereas in the second case, it also estimates...
A posteriori error estimates for vertex centered finite volume approximations of convection-diffusion-reaction equations
This paper is devoted to the study of a posteriori error estimates for the scalar nonlinear convection-diffusion-reaction equation . The estimates for the error between the exact solution and an upwind finite volume approximation to the solution are derived in the -norm, independent of the diffusion parameter . The resulting a posteriori error estimate is used to define an grid adaptive solution algorithm for the finite volume scheme. Finally numerical experiments underline the applicability...
A posteriori error estimates for vertex centered finite volume approximations of convection-diffusion-reaction equations
This paper is devoted to the study of a posteriori error estimates for the scalar nonlinear convection-diffusion-reaction equation . The estimates for the error between the exact solution and an upwind finite volume approximation to the solution are derived in the L1-norm, independent of the diffusion parameter D. The resulting a posteriori error estimate is used to define an grid adaptive solution algorithm for the finite volume scheme. Finally numerical experiments underline the applicability...
A posteriori error estimates of the discontinuous Galerkin method for the heat conduction equation
A posteriori error estimates with post-processing for nonconforming finite elements
For a nonconforming finite element approximation of an elliptic model problem, we propose a posteriori error estimates in the energy norm which use as an additive term the “post-processing error” between the original nonconforming finite element solution and an easy computable conforming approximation of that solution. Thus, for the error analysis, the existing theory from the conforming case can be used together with some simple additional arguments. As an essential point, the property is exploited...
A posteriori Error Estimates with Post-Processing for Nonconforming Finite Elements
For a nonconforming finite element approximation of an elliptic model problem, we propose a posteriori error estimates in the energy norm which use as an additive term the “post-processing error” between the original nonconforming finite element solution and an easy computable conforming approximation of that solution. Thus, for the error analysis, the existing theory from the conforming case can be used together with some simple additional arguments. As an essential point, the property is...
A Posteriori Error Estimation for Reduced Order Solutions of Parametrized Parabolic Optimal Control Problems
We consider the efficient and reliable solution of linear-quadratic optimal control problems governed by parametrized parabolic partial differential equations. To this end, we employ the reduced basis method as a low-dimensional surrogate model to solve the optimal control problem and develop a posteriori error estimation procedures that provide rigorous bounds for the error in the optimal control and the associated cost functional. We show that our approach can be applied to problems involving...
A posteriori error estimation for reduced-basis approximation of parametrized elliptic coercive partial differential equations : “convex inverse” bound conditioners
We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic coercive partial differential equations with affine parameter dependence. The essential components are (i) (provably) rapidly convergent global reduced-basis approximations – Galerkin projection onto a space spanned by solutions of the governing partial differential equation at selected points in parameter space; (ii) a posteriori error estimation – relaxations of the error-residual equation...
A Posteriori Error Estimation for Reduced-Basis Approximation of Parametrized Elliptic Coercive Partial Differential Equations: “Convex Inverse” Bound Conditioners
We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic coercive partial differential equations with affine parameter dependence. The essential components are (i ) (provably) rapidly convergent global reduced-basis approximations – Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii ) a posteriori error estimation – relaxations of the error-residual equation...
A posteriori error estimation for semilinear parabolic optimal control problems with application to model reduction by POD
We consider the following problem of error estimation for the optimal control of nonlinear parabolic partial differential equations: let an arbitrary admissible control function be given. How far is it from the next locally optimal control? Under natural assumptions including a second-order sufficient optimality condition for the (unknown) locally optimal control, we estimate the distance between the two controls. To do this, we need some information on the lowest eigenvalue of the reduced Hessian....
A posteriori error estimation with finite element methods of lines for one-dimensional parabolic systems.
A posteriori error estimators for nonconforming finite element methods
A Posteriori Error Estimators for the Stokes Equations.
A posteriori error estimators in the finite element method.
A posteriori estimates for the Cahn–Hilliard equation with obstacle free energy
We derive a posteriori estimates for a discretization in space of the standard Cahn–Hilliard equation with a double obstacle free energy. The derived estimates are robust and efficient, and in practice are combined with a heuristic time step adaptation. We present numerical experiments in two and three space dimensions and compare our method with an existing heuristic spatial mesh adaptation algorithm.
A posteriori estimations of approximate solutions for some types of boundary value problems
A posteriori upper and lower error bound of the high-order discontinuous Galerkin method for the heat conduction equation
We deal with the numerical solution of the nonstationary heat conduction equation with mixed Dirichlet/Neumann boundary conditions. The backward Euler method is employed for the time discretization and the interior penalty discontinuous Galerkin method for the space discretization. Assuming shape regularity, local quasi-uniformity, and transition conditions, we derive both a posteriori upper and lower error bounds. The analysis is based on the Helmholtz decomposition, the averaging interpolation...