Error estimates for fully discrete generalized FEMs with locally optimal spectral approximations

TL;DR

This paper analyzes error estimates for fully discrete generalized finite element methods using locally optimal spectral approximations, demonstrating nearly exponential convergence rates and proposing an efficient eigenproblem solver.

Contribution

It introduces a new error analysis for discrete GFEM with optimal local spaces and proposes an improved eigenproblem solution method.

Findings

Error bounds converge as mesh size decreases, approaching continuous GFEM error.

Local approximation errors decay nearly exponentially with space dimension.

Numerical experiments confirm theoretical convergence and efficiency.

Abstract

This paper is concerned with error estimates of the fully discrete generalized finite element method (GFEM) with optimal local approximation spaces for solving elliptic problems with heterogeneous coefficients. The local approximation spaces are constructed using eigenvectors of local eigenvalue problems solved by the finite element method on some sufficiently fine mesh with mesh size $h$. The error bound of the discrete GFEM approximation is proved to converge as $h\rightarrow 0$ towards that of the continuous GFEM approximation, which was shown to decay nearly exponentially in previous works. Moreover, even for fixed mesh size $h$, a nearly exponential rate of convergence of the local approximation errors with respect to the dimension of the local spaces is established. An efficient and accurate method for solving the discrete eigenvalue problems is proposed by incorporating the discrete $A$-harmonic constraint directly into the eigensolver. Numerical experiments are carried out to confirm the theoretical results and to demonstrate the effectiveness of the method.