The adjoint double layer potential on smooth surfaces in R3 and the Neumann problem

• Published in: Advances in computational mathematics Vol. 50; no. 3
• Main Authors: Beale, J. Thomas, Storm, Michael, Tlupova, Svetlana
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2024; Springer Nature B.V
• Subjects: Boundary value problems; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Electric double layer; Green's functions; Integral equations; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Neumann problem; Regularization; Spherical coordinates; Visualization; 65D30; 65R20; Layer potential; Singular integral; 31B10; Neumann boundary condition; Boundary integral method; Laplace equation; Adjoint double layer; 35J25
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-024-10111-0

We present a simple yet accurate method to compute the adjoint double layer potential, which is used to solve the Neumann boundary value problem for Laplace’s equation in three dimensions. An expansion in curvilinear coordinates leads us to modify the expression for the adjoint double layer so that the singularity is reduced when evaluating the integral on the surface. Then, to regularize the integral, we multiply the Green’s function by a radial function with length parameter δ chosen so that the product is smooth. We show that a natural regularization has error O ( δ 3 ) , and a simple modification improves the error to O ( δ 5 ) . The integral is evaluated numerically without the need of special coordinates. We use this treatment of the adjoint double layer to solve the classical integral equation for the interior Neumann problem, altered to account for the solvability condition, and evaluate the solution on the boundary. Choosing δ = c h 4 / 5 , we find about O ( h 4 ) convergence in our examples, where h is the spacing in a background grid.