Boundary Value Correction for the Boundary Element Method

We develop a boundary element method for the Laplace equation with Dirichlet boundary conditions in a bounded three-dimensional domain. The method uses flat panels of maximum diameter $h$ on an interior surrogate boundary and corrects the imposed boundary values by a first-order Taylor expansion along the panel normals. The correction uses the normal flux already present in the Calderón formulation and introduces no additional unknown field. Under uniform assumptions on the surrogate geometry and boundary operators, the penalised form is coercive when $\gamma {\delta }_h\le c_0<4$, where $\gamma$ is the penalty parameter and ${\delta }_h$ is the maximum normal offset. The error estimate separates trace approximation from a geometry term of size ${\gamma }^{1\mathrm{/}2}{\delta }_h^2\mathrm{\parallel }u{\mathrm{\parallel }}_{H^3(\mathrm{\Omega })}$. With $\gamma \simeq h^{-1}$, the sufficient condition ${\delta }_h\lesssim h^{(k+1)\mathrm{/}2}$ yields an error of order $h^{k+1\mathrm{/}2}$ for $P_k\mathrm{/}{P}_{k-1}$ Dirichlet/flux spaces under the stated regularity and uniform trace bounds. Thus an offset of order $h^2$ supports the estimate through cubic Dirichlet approximation. We analyse two penalty variants and present numerical experiments on a sphere and an ellipsoid to examine convergence and sensitivity to geometry, penalty scaling and quadrature.