New L2 error estimates for weighted L2 projections of H1 functions are controlled by the H1 semi-norm except under highly irregular weights such as checkerboard patterns.
Xu, Counterexamples concerning a weighted L2 projection, Mathematics of Computation, 57(1991), 563-568
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New error estimates of the weighted $L^2$ projections
New L2 error estimates for weighted L2 projections of H1 functions are controlled by the H1 semi-norm except under highly irregular weights such as checkerboard patterns.