New error estimates of the weighted $L^2$ projections

TL;DR

This paper derives sharper $L^2$ error estimates for weighted $L^2$ projections, showing they can be controlled by the $H^1$ semi-norm except in highly irregular weights, aiding PDE numerical methods.

Contribution

The paper establishes new $L^2$ error bounds for weighted projections under general weights, improving understanding of their approximation properties.

Findings

Error estimates depend on weight regularity

Errors are controlled by $H^1$ semi-norm for regular weights

Irregular weights like checkerboard patterns cause exceptions

Abstract

It is known that the weighted $L^2$ projection operator exhibits approximation properties different from those of the classical $L^2$ projection, in the sense that the $L^2$ error of the weighted $L^2$ projection of an $H^1$ function generally cannot be bounded by the $H^1$ semi-norm of the function. In this paper, we establish sharper $L^2$ error estimates for the weighted $L^2$ projection of an $H^1$ function under general weight distributions. These new estimates show that the $L^2$ errors of the weighted $L^2$ projection can be controlled by the $H^1$ semi-norm of the function, except when the weight distribution is highly irregular, such as those resembling a ``checkerboard" pattern. These results can be applied to more refined analyses of domain decomposition methods and multigrid methods for certain partial differential equations with large jump coefficients.