# Adaptive FEM for Helmholtz Equation with Large Wave Number

@article{Duan2021AdaptiveFF, title={Adaptive FEM for Helmholtz Equation with Large Wave Number}, author={Songyao Duan and Haijun Wu}, journal={ArXiv}, year={2021}, volume={abs/2110.11939} }

A posteriori upper and lower bounds are derived for the linear finite element method (FEM) for the Helmholtz equation with large wave number. It is proved rigorously that the standard residual type error estimator seriously underestimates the true error of the FE solution for the mesh size h in the preasymptotic regime, which is first observed by Babuška, et al. for an one dimensional problem. By establishing an equivalence relationship between the error estimators for the FE solution and the…

## References

SHOWING 1-10 OF 49 REFERENCES

Finite element solution of the Helmholtz equation with high wave number Part I: The h-version of the FEM☆

- Mathematics
- 1995

Abstract The paper addresses the properties of finite element solutions for the Helmholtz equation. The h-version of the finite element method with piecewise linear approximation is applied to a…

Continuous Interior Penalty Finite Element Method for Helmholtz Equation with High Wave Number: One Dimensional Analysis

- Mathematics
- 2012

This paper addresses the properties of Continuous Interior Penalty (CIP) finite element solutions for the Helmholtz equation. The $h$-version of the CIP finite element method with piecewise linear…

A Posteriori Error Estimation for Highly Indefinite Helmholtz Problems

- Mathematics, Computer ScienceComput. Methods Appl. Math.
- 2013

This paper will introduce an a posteriori error estimator for this problem and prove its reliability and efficiency and emphasize that the constants in these estimates become independent of the, possibly, high wavenumber provided the aforementioned resolution condition for stability is satisfied.

Preasymptotic Error Analysis of Higher Order FEM and CIP-FEM for Helmholtz Equation with High Wave Number

- Physics, MathematicsSIAM J. Numer. Anal.
- 2015

It is shown that if the pollution error of the CIP-FEM is sufficiently small, then the pollution errors of both methods in $H^1$-norm are bounded by $O(k^{2p+1}h 2p})$, which coincides with the phaseerror of the FEM obtained by existent dispersion analyses on Cartesian grids.

Pre-asymptotic error analysis of CIP-FEM and FEM for the Helmholtz equation with high wave number. Part I: linear version

- Mathematics
- 2014

On the derivation of guaranteed and p-robust a posteriori error estimates for the Helmholtz equation

- Computer Science, MathematicsNumerische Mathematik
- 2021

A novel a posteriori error estimator for conforming finite element discretizations of two-and three-dimensional Helmholtz problems based on an equilibrated flux that is computed through the solve of patchwise mixed finite element problems is proposed.

Superconvergence analysis of linear FEM based on polynomial preserving recovery for Helmholtz equation with high wave number

- Computer Science, MathematicsJ. Comput. Appl. Math.
- 2020

The superconvergence result is obtained which says that the PPR improves only the interpolation error but keeps the pollution error unchanged, and it is demonstrated that thePPR combining with the continuous interior penalty technique is much effective for improving both the interpolations error and the pollutionerror.

Convergence analysis of an adaptive continuous interior penalty finite element method for the Helmholtz equation

- Mathematics
- 2015

Abstract We are concerned with an adaptive continuous interior penalty finite element method for the Helmholtz equation. A convergence result with explicit constraints on the wave number k and the…

Wavenumber Explicit Convergence Analysis for Galerkin Discretizations of the Helmholtz Equation

- Computer Science, MathematicsSIAM J. Numer. Anal.
- 2011

A stability and convergence theory for a class of highly indefinite elliptic boundary value problems (bvps) by considering the Helmholtz equation at high wavenumber $k$ as the authors' model problem and it is shown that quasi optimality is obtained under the conditions that $kh/ p$ is sufficiently small and the polynomial degree $p$ is at least O(log $k) .

Convergence analysis of an adaptive interior penalty discontinuous Galerkin method for the Helmholtz equation

- Mathematics
- 2013

In this thesis, we are mainly concerned with the numerical solution of the two dimensional Helmholtz equation by an adaptive Interior Penalty Discontinuous Galerkin (IPDG) method based on adaptively…