A born-weight boundary element method for multiphysics

Classical boundary element methods lose efficiency where the boundary field is oscillatory, multi-scale or evanescent, for which polynomial interpolation demands a fine mesh. This paper presents the Born-weight boundary element method (BWBEM): the fixed polynomial shape function is replaced by a non-negative, physics-adapted partition of unity , applied unchanged across the boundary-integral operators. It is a drop-in reparametrisation — kernels and unknowns untouched — and an operator-level coupling that embeds interface continuity follows from the same weights. The quantum vocabulary is a motivating analogy only; every construction stands in ordinary terms. The distinctive property is near-spectral accuracy with shape preservation: for smooth data the Born weights converge super-algebraically and, a convex combination of nodal coefficients, add no inter-nodal oscillation, where a spectral basis of equal accuracy rings (Gibbs). The paper is candid: accuracy alone gives no advantage over the spectral basis, which matches or beats it and does not floor, and on coarse meshes the enrichment can be less accurate than linear elements. In a solve, a bounded-concentration test shows the bound respected to round-off once resolved, and violated far less than by the spectral basis otherwise. The decisive gain, for singular fields, is left to a companion study. It is validated against exact references throughout—the three scalar regimes, Stokes drag, acoustic scattering, quantum tunnelling, the coupled interface, and a non-circular geometry with real singular quadrature. The dense operator, of classical-BEM complexity, reaches a target accuracy at far fewer unknowns; sub-quadratic scaling under compression is future work, not claimed.