Related papers: Worst-case multi-objective error estimation and ad…
A general framework for goal-oriented a posteriori error estimation for finite volume methods is presented. The framework does not rely on recasting finite volume methods as special cases of finite element methods, but instead directly…
The convergence analysis for least-squares finite element methods led to various adaptive mesh-refinement strategies: Collective marking algorithms driven by the built-in a posteriori error estimator or an alternative explicit…
In this work, we further develop multigoal-oriented a posteriori error estimation with two objectives in mind. First, we formulate goal-oriented mesh adaptivity for multiple functionals of interest for nonlinear problems in which both the…
Recovery type a posteriori error estimators are popular, particularly in the engineering community, for their computationally inexpensive, easy to implement, and generally asymptotically exactness. Unlike the residual type error estimators,…
The state-of-the art proof of a global inf-sup condition on mixed finite element schemes does not allow for an analysis of truly indefinite, second-order linear elliptic PDEs. This paper, therefore, first analyses a nonconforming finite…
We introduce an adaptive superconvergent finite element method for a class of mixed formulations to solve partial differential equations involving a diffusion term. It combines a superconvergent postprocessing technique for the primal…
A multilevel adaptive refinement strategy for solving linear elliptic partial differential equations with random data is recalled in this work. The strategy extends the a posteriori error estimation framework introduced by Guignard and…
In this paper we develop two goal-oriented adaptive strategies for a posteriori error estimation within the generalized multiscale finite element framework. In this methodology, one seeks to determine the number of multiscale basis…
We consider finite element discretizations of Maxwell's equations coupled with a non-local hydrodynamic Drude model that accurately accounts for electron motions in metallic nanostructures. Specifically, we focus on a posteriori error…
In a general setting, we study a posteriori estimates used in finite element analysis to measure the error between a solution and its approximation. The latter is not necessarily generated by a finite element method. We show that the error…
Deep learning has shown successful application in visual recognition and certain artificial intelligence tasks. Deep learning is also considered as a powerful tool with high flexibility to approximate functions. In the present work,…
This paper aims to develop an efficient adaptive finite element method for the second-order elliptic problem. Although the theory for adaptive finite element methods based on residual-type a posteriori error estimator and bisection…
The a posteriori error estimator using the least-squares functional can be used for adaptive mesh refinement and error control even if the numerical approximations are not obtained from the corresponding least-squares method. This suggests…
The convergence of an adaptive mixed finite element method for general second order linear elliptic problems defined on simply connected bounded polygonal domains is analyzed in this paper. The main difficulties in the analysis are posed by…
Adaptive atomistic/continuum (a/c) coupling method is an important method for the simulation of material and atomistic systems with defects to achieve the balance of accuracy and efficiency. Residual based a posteriori error estimator is…
This article is a review on basic concepts and tools devoted to a posteriori error estimation for problems solved with the Finite Element Method. For the sake of simplicity and clarity, we mostly focus on linear elliptic diffusion problems,…
This paper reviews the state of the art and discusses very recent mathematical developments in the field of adaptive boundary element methods. This includes an overview of available a posteriori error estimates as well as a state-of-the-art…
In this work, we apply multi-goal oriented error estimation to the finite element method. In particular, we use the dual weighted residual method and apply it to a model problem. This model problem consist of locally different coercive…
We develop all of the components needed to construct an adaptive finite element code that can be used to approximate fractional partial differential equations, on non-trivial domains in $d\geq 1$ dimensions. Our main approach consists of…
In this paper, we study an adaptive finite element method for multiple eigenvalue problems of a class of second order elliptic equations. By using some eigenspace approximation technology and its crucial property which is also presented in…