A remark on the numerical integration of harmonic functions in the plane

We show that certain domains Ω⊂R2Ω⊂R2 have the following property: there is a sequence of points (xi)i=1∞ in ΩΩ with nonnegative weights (ai)i=1∞ such that for all harmonic functions u:R2→Ru:R2→R and all N≥1N≥1 we have |∫Ωu(x)dx−∑i=1Naiu(xi)|≤CΩ‖u‖L∞(Ω)N0.53, where CΩCΩ depends only on ΩΩ. We emphasize that the points (xi)(xi) and the weights (ai)(ai) do not depend on uu. This improves on the (probabilistic) Monte-Carlo bound ‖u‖L2(Ω)/N0.5‖u‖L2(Ω)/N0.5without involving any sort of control on the oscillation of the function (which is classically done via the size of derivatives or the total variation). We do not know which decay rate is optimal.