New error estimates of the weighted $L^2$ projections
Provides theoretical guarantees for weighted L^2 projections, enabling more refined analyses of domain decomposition and multigrid methods for PDEs with large jump coefficients.
The paper establishes sharper L^2 error estimates for weighted L^2 projections of H^1 functions, showing that errors can be controlled by the H^1 semi-norm except for highly irregular weight distributions like checkerboard patterns.
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.