Convergence and Complexity of an Adaptive Planewave Method for Eigenvalue Computations

TL;DR

This paper develops an adaptive planewave method for eigenvalue problems in elliptic PDEs, providing error estimation, convergence analysis, and demonstrating quasi-optimal complexity.

Contribution

It introduces an a posteriori error estimator and proves linear convergence and quasi-optimal complexity of the adaptive planewave method.

Findings

Proposed an effective a posteriori error estimator.

Proved linear convergence rate of the adaptive method.

Established quasi-optimal complexity of the approach.

Abstract

In this paper, we study the adaptive planewave discretization for a cluster of eigenvalues of second-order elliptic partial differential equations. We first design an a posteriori error estimator and prove both the upper and lower bounds. Based on the a posteriori error estimator, we propose an adaptive planewave method. We then prove that the adaptive planewave approximations have the linear convergence rate and quasi-optimal complexity.