Multigoal-oriented adaptive finite element method with convergence rates

TL;DR

This paper develops a goal-oriented adaptive finite element method for symmetric linear elliptic PDEs that handles multiple goal functionals simultaneously, providing convergence analysis and numerical validation.

Contribution

It introduces a novel adaptive FEM framework for multiple goal functionals with proven convergence and optimal rates, unlike previous approaches.

Findings

Algorithm guarantees convergence rates

Numerical experiments confirm theoretical results

Handles multiple goal functionals efficiently

Abstract

We formulate and analyze a goal-oriented adaptive finite element method for a symmetric linear elliptic partial differential equation (PDE) that can simultaneously deal with multiple linear goal functionals. In each step of the algorithm, only two linear finite element systems have to be solved. Moreover, all finite element solutions are computed with respect to the same discrete space, while the underlying triangulations are adapted to resolve all inherent singularities simultaneously. Unlike available results for such a setting in the literature, we give a thorough convergence analysis and verify that our algorithm guarantees, in an appropriate sense, even optimal convergence rates. Numerical experiments underline the derived theoretical results.