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We derive a computable a posteriori error estimator for the $\alpha$-harmonic extension problem, which localizes the fractional powers of elliptic operators supplemented with Dirichlet boundary conditions. Our a posteriori error estimator…

Numerical Analysis · Mathematics 2015-05-20 Long Chen , Ricardo H. Nochetto , Enrique Otárola , Abner J. Salgado

In this paper, a residual-type a posteriori error estimator is proposed and analyzed for a modified weak Galerkin finite element method solving linear elasticity problems. The estimator is proven to be both reliable and efficient because it…

Numerical Analysis · Mathematics 2023-02-21 Liu Chunmei , Zhong Liuqiang , Xie Yingying Xie , Zhou Liping

We derive a posteriori error estimates in the $L_\infty((0,T];L_\infty(\Omega))$ norm for approximations of solutions to linear para bolic equations. Using the elliptic reconstruction technique introduced by Makridakis and Nochetto and heat…

Numerical Analysis · Mathematics 2011-04-06 Alan Demlow , Omar Lakkis , Charalambos Makridakis

In this work we derive a posteriori error estimates for the convection-diffusion-reaction equation coupled with the Darcy-Forchheimer problem by a nonlinear external source depending on the concentration of the fluid. We introduce the…

Numerical Analysis · Mathematics 2022-12-27 Toni Sayah , Georges Semaan , Faouzi Triki

This paper concerns a posteriori error analysis for the streamline diffusion (SD) finite element method for the one and one-half dimensional relativistic Vlasov-Maxwell system. The SD scheme yields a weak formulation, that corresponds to an…

Numerical Analysis · Mathematics 2016-12-23 Mohammad Asadzadeh , Christoffer Standar

In this paper, we propose a new family of H(curl^2)-conforming elements for the quad-curl eigenvalue problem in 2D. The accuracy of this family is one order higher than that in [32]. We prove a priori and a posteriori error estimates. The a…

Numerical Analysis · Mathematics 2020-07-06 Lixiu Wang , Qian Zhang , Jiguang Sun , Zhimin Zhang

In the present work, we derive functional upper bounds for the potential error arising from finite-element boundary-element coupling formulations for a nonlinear Poisson-type transmission problem. The proposed a posteriori error estimates…

Numerical Analysis · Mathematics 2026-02-17 Alexander Freiszlinger , Dirk Pauly , Dirk Praetorius , Michael Schomburg

A posteriori estimates give bounds on the error between the unknown solution of a partial differential equation and its numerical approximation. We present here the methodology based on H1-conforming potential and H(div)-conforming…

Numerical Analysis · Mathematics 2025-05-30 Martin Vohralík , Soleiman Yousef

We combine a systematic approach for deriving general a posteriori error estimates for convex minimization problems based on convex duality relations with a recently derived generalized Marini formula. The a posteriori error estimates are…

Numerical Analysis · Mathematics 2022-04-25 Sören Bartels , Alex Kaltenbach

We develop an \textit{a posteriori} error analysis for a numerical estimate of the time at which a functional of the solution to a partial differential equation (PDE) first achieves a threshold value on a given time interval. This quantity…

Numerical Analysis · Mathematics 2022-06-09 Jehanzeb Chaudhry , Don Estep , Trevor Giannini , Zachary Stevens , Simon Tavener

We derive $H_{\text{curl}}$-error estimates and improved $L^2$-error estimates for the Maxwell equations approximated using edge finite elements. These estimates only invoke the expected regularity pickup of the exact solution in the scale…

Numerical Analysis · Mathematics 2017-10-17 Alexandre Ern , Jean-Luc Guermond

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…

Numerical Analysis · Mathematics 2017-04-24 Alejandro Allendes , Enrique Otarola , Richard Rankin

We devise an a posteriori error estimator for an affine optimal control problem subject to a semilinear elliptic PDE and control constraints. To approximate the problem, we consider a semidiscrete scheme based on the variational…

Optimization and Control · Mathematics 2025-05-08 Francisco Fuica , Nicolai Jork

We perform the a posteriori error analysis of residual type of a transmission problem with sign changing coefficients. According to [6] if the contrast is large enough, the continuous problem can be transformed into a coercive one. We…

Numerical Analysis · Mathematics 2010-09-17 Serge Nicaise , Juliette Venel

We consider second-order PDE problems set in unbounded domains and discretized by Lagrange finite elements on a finite mesh, thus introducing an artificial boundary in the discretization. Specifically, we consider the reaction diffusion…

Numerical Analysis · Mathematics 2025-03-31 T. Chaumont-Frelet

A posteriori error estimates are an important tool to bound discretization errors in terms of computable quantities avoiding regularity conditions that are often difficult to establish. For non-linear and non-differentiable problems,…

Numerical Analysis · Mathematics 2024-06-12 Sören Bartels , Alex Kaltenbach

We propose a novel a posteriori error estimator for the N\'ed\'elec finite element discretization of time-harmonic Maxwell's equations. After the approximation of the electric field is computed, we propose a fully localized algorithm to…

Numerical Analysis · Mathematics 2024-02-28 T. Chaumont-Frelet

A posteriori error estimators for the symmetric mixed finite element methods for linear elasticity problems of Dirichlet and mixed boundary conditions are proposed. Stability and efficiency of the estimators are proved. Finally, we provide…

Numerical Analysis · Mathematics 2017-05-12 Long Chen , Jun Hu , Xuehai Huang , Hongying Man

In this work, we propose an a pointwise a posteriori error estimator for conforming finite element approximations of eigenfunctions corresponding to multiple and clustered eigenvalues of elliptic operators. It is proven that the pointwise a…

Numerical Analysis · Mathematics 2025-11-12 Zhenglei Li , Qigang Liang , Xuejun Xu

A posteriori estimates for mixed finite element discretizations of the Navier-Stokes equations are derived. We show that the task of estimating the error in the evolutionary Navier-Stokes equations can be reduced to the estimation of the…

Numerical Analysis · Mathematics 2016-12-23 Javier de Frutos , Bosco García-Archilla , Julia Novo