Worst-case multi-objective error estimation and adaptivity

Abstract

This paper introduces a new computational methodology for determining aposteriori multi-objective error estimates for finite-element approximations, and for constructing corresponding (quasi-)optimal adaptive refinements of finiteelement spaces. As opposed to the classical goal-oriented approaches, which consider only a single objective functional, the presented methodology applies to general closed convex subsets of the dual space and constructs a worst-case error estimate of the finite-element approximation error. This worst-case multiobjective error estimate conforms to a dual-weighted residual, in which the dual solution is associated with an approximate supporting functional of the objective set at the approximation error. We regard both standard approximation errors and data-incompatibility errors associated with incompatibility of boundary data with the trace of the finite-element space. Numerical experiments are presented to demonstrate the efficacy of applying the proposed worst-case multiobjective error in adaptive refinement procedures.

Singular Nature of the Elastocapillary Ridge

“…In goal-adaptive methods, the finite-element approximation is locally refined on the basis of an a posteriori error estimate, in such a manner that an optimal approximation to a predefined quantity of interest (the goal/) is obtained. Goal-adaptive finite-element methods generally proceed according to the solve → estimate → mark → refine (SEMR) process [44,45]. The SEMR process starts by solving a finite-element approximation on a coarse mesh.…”

A Posteriori Error Estimation and Adaptivity for Nonlinear Parabolic Equations using IMEX-Galerkin Discretization of Primal and Dual Equations

“…For spatial adaptivity we consider hierarchical mesh-refinement indicators as explained in [29,Section 4.2]. In addition, let us note that instead of a traditional element-wise marking strategy, we use the function-support marking strategy introduced in [23] (see also [26,31]).…”

“…, |E k M |}). The addition of the basis function on selected nodes is performed using hierarchical refinement for finite element methods [22,23,26,31]. Moreover, instead of projection, we introduce a common refinement to transfer the solution from one mesh to another without loss of accuracy in any quadrature approximations.…”

Goal-oriented a posteriori error estimation for conforming and nonconforming approximations with inexact solvers

“…The dual weighted residual (DWR) method promoted by Becker and Rannacher [5], see also [3,7,22,24,33], the general framework by Prodhumme and Oden [36,40], the approach of Maday and Patera [31], multi-objective error estimation in [16,23,46], enhanced least-squares finite element methods by Chaudhry et al [8], or the constitutive relation error (CRE) approach of Ladevèze et al [28,30] and Rey et al [42][43][44], see also the references therein, are very popular approaches to goal-oriented error estimation; this can also be built in the discretization scheme as in Kergrene et al [27]. The obtained bounds are, however, often not guaranteed in the sense of yielding a fully computable number that is rigorously greater than or equal to the goal error.…”