Boundary Value Correction for the Boundary Element Method
Abstract
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 γδh≤ c0<4, where γ is the penalty parameter and δh is the maximum normal offset. The error estimate separates trace approximation from a geometry term of size γ1/2δh2|u|H3(Ω). With γ≃ h−1, the sufficient condition δh≲ h(k+1)/2 yields an error of order hk+1/2 for Pk/Pk−1 Dirichlet/flux spaces under the stated regularity and uniform trace bounds. Thus an offset of order h2 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.
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BibTeX
@misc{burman2026boundary,
title = {{Boundary Value Correction for the Boundary Element Method}},
author = {Erik Burman and Peter Hansbo and Mats G. Larson},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31555},
eprint = {2609.31555},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31555},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Boundary Value Correction for the Boundary Element Method
AU - Erik Burman
AU - Peter Hansbo
AU - Mats G. Larson
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31555
UR - https://arxiv.org/abs/2609.31555
AB - 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 γδ_h≤ c_0<4, where γ is the penalty parameter and δ_h is the maximum normal offset. The error estimate separates trace approximation from a geometry term of size γ^1/2δ_h^2|u|_H^3(Ω). With γ≃ h^−1, the sufficient condition δ_h≲ h^(k+1)/2 yields an error of order h^k+1/2 for P_k/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.
ER -