A posteriori error estimate of approximate solutions to a mildly nonlinear elliptic boundary value problem
A posteriori error estimates for a discontinuous Galerkin approximation of Steklov eigenvalue problems
We derive a residual-based a posteriori error estimator for a discontinuous Galerkin approximation of the Steklov eigenvalue problem. Moreover, we prove the reliability and efficiency of the error estimator. Numerical results are provided to verify our theoretical findings.
A posteriori error estimates for a nonconforming finite element discretization of the heat equation
The paper presents an a posteriori error estimator for a (piecewise linear) nonconforming finite element approximation of the heat equation in , or 3, using backward Euler’s scheme. For this discretization, we derive a residual indicator, which use a spatial residual indicator based on the jumps of normal and tangential derivatives of the nonconforming approximation and a time residual indicator based on the jump of broken gradients at each time step. Lower and upper bounds form the main results...
A posteriori error estimates for a nonconforming finite element discretization of the heat equation
The paper presents an a posteriori error estimator for a (piecewise linear) nonconforming finite element approximation of the heat equation in , d=2 or 3, using backward Euler's scheme. For this discretization, we derive a residual indicator, which use a spatial residual indicator based on the jumps of normal and tangential derivatives of the nonconforming approximation and a time residual indicator based on the jump of broken gradients at each time step. Lower and upper bounds form the main...
A posteriori error estimates for elliptic problems with Dirac measure terms in weighted spaces
In this article we develop a posteriori error estimates for second order linear elliptic problems with point sources in two- and three-dimensional domains. We prove a global upper bound and a local lower bound for the error measured in a weighted Sobolev space. The weight considered is a (positive) power of the distance to the support of the Dirac delta source term, and belongs to the Muckenhoupt’s class A2. The theory hinges on local approximation properties of either Clément or Scott–Zhang interpolation...
A Posteriori Error Estimates for Finite Volume Approximations
We present new a posteriori error estimates for the finite volume approximations of elliptic problems. They are obtained by applying functional a posteriori error estimates to natural extensions of the approximate solution and its flux computed by the finite volume method. The estimates give guaranteed upper bounds for the errors in terms of the primal (energy) norm, dual norm (for fluxes), and also in terms of the combined primal-dual norms. It is shown that the estimates provide sharp upper and...
A posteriori error estimates for linear exterior problems via mixed-FEM and DtN mappings
In this paper we combine the dual-mixed finite element method with a Dirichlet-to-Neumann mapping (given in terms of a boundary integral operator) to solve linear exterior transmission problems in the plane. As a model we consider a second order elliptic equation in divergence form coupled with the Laplace equation in the exterior unbounded region. We show that the resulting mixed variational formulation and an associated discrete scheme using Raviart-Thomas spaces are well posed, and derive the...
A posteriori error estimates for nonlinear problems. -estimates for finite element discretizations of elliptic equations
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 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-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 estimators for nonconforming finite element methods
A Posteriori Error Estimators for the Stokes Equations.
A posteriori error estimators in the finite element method.
A Primal Hybrid Finite Element Method for Quasilinear Second Order Elliptic Problems.
A priori and a posteriori error bounds for a nonconforming linear finite element approximation of a non-newtonian flow
A priori convergence of the greedy algorithm for the parametrized reduced basis method
The convergence and efficiency of the reduced basis method used for the approximation of the solutions to a class of problems written as a parametrized PDE depends heavily on the choice of the elements that constitute the “reduced basis”. The purpose of this paper is to analyze the a priori convergence for one of the approaches used for the selection of these elements, the greedy algorithm. Under natural hypothesis on the set of all solutions to the problem obtained when the parameter varies, we...