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

Qiya Hu; Yuhan Luo

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arXiv · 2605.05637

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

Qiya Hu·Yuhan Luo

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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.

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Qiya Hu, Yuhan Luo. 2026-05-07. New error estimates of the weighted $L^2$ projections. https://arxiv.org/abs/2605.05637

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