Plain convergence of adaptive algorithms without exploiting reliability and efficiency

  title={Plain convergence of adaptive algorithms without exploiting reliability and efficiency},
  author={Gregor Gantner and Dirk Praetorius},
We consider $h$-adaptive algorithms in the context of the finite element method and the boundary element method. Under quite general assumptions on the building blocks SOLVE, ESTIMATE, MARK and REFINE of such algorithms we prove plain convergence in the sense that the adaptive algorithm drives the underlying a posteriori error estimator to zero. Unlike available results in the literature, our analysis avoids the use of any reliability and efficiency estimate but relies only on structural… 

Plain convergence of goal-oriented adaptive FEM

We discuss goal-oriented adaptivity in the frame of conforming finite element methods and plain convergence of the related a posteriori error estimator for different general marking strategies. We

Adaptive Quasi-Monte Carlo Finite Element Methods for Parametric Elliptic PDEs

  • M. Longo
  • Computer Science, Mathematics
    Journal of Scientific Computing
  • 2022
A class of deterministic quasi-Monte Carlo integration rules derived from Polynomial lattices is performed that allows to control a-posteriori the integration error without querying the governing PDE and does not incur the curse of dimensionality.



Estimator reduction and convergence of adaptive BEM


We consider the approximate solution with adaptive finite elements of a class of linear boundary value problems, which includes problems of "saddle point" type. For the adaptive algorithm we assume

A short note on plain convergence of adaptive least-squares finite element methods

Quasi-optimal convergence rate for an adaptive method for the integral fractional Laplacian

A reliable weighted residual a posteriori error estimator for the discretization of the integral fractional Laplacian based on piecewise linear functions is presented and it is proved optimal convergence rates for an $h$-adaptive algorithm driven by this error estimators.

Convergence analysis of a conforming adaptive finite element method for an obstacle problem

Surprisingly, the adaptive mesh-refinement algorithm is the same as in the unconstrained case of a linear PDE—in fact, there is no modification near the discrete free boundary necessary for R-linear convergence.

Adaptive FEM with Optimal Convergence Rates for a Certain Class of Nonsymmetric and Possibly Nonlinear Problems

This work analyzes adaptive mesh-refining algorithms for conforming finite element discretizations of certain nonlinear second-order partial differential equations and proves convergence even with optimal algebraic convergence rates.

Quasi-optimal Convergence Rate for an Adaptive Boundary Element Method

It is proved that adaptive mesh refinement is superior to uniform mesh refinement and convergence of an $h$-adaptive algorithm that is driven by a weighted residual error estimator.

Adaptive boundary element methods with convergence rates

The main ingredients of the proof that constitute new findings are some results on a posteriori error estimates for boundary element methods, and an inverse-type inequality involving boundary integral operators on locally refined finite element spaces.

Axioms of adaptivity