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

Abstract

We show that certain domains Ω⊂R2 have the following property: there is a sequence of points (xi)i=1∞ in Ω with nonnegative weights (ai)i=1∞ such that for all harmonic functions u:R2→R and all N≥1 we have |∫Ωu(x)dx−∑i=1Naiu(xi)|≤CΩ‖u‖L∞(Ω)N0.53, where CΩ depends only on Ω. We emphasize that the points (xi) and the weights (ai) do not depend on u. This improves on the (probabilistic) Monte-Carlo bound ‖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.